Best linear algebra books according to redditors
We found 297 Reddit comments discussing the best linear algebra books. We ranked the 80 resulting products by number of redditors who mentioned them. Here are the top 20.
We found 297 Reddit comments discussing the best linear algebra books. We ranked the 80 resulting products by number of redditors who mentioned them. Here are the top 20.
Here's my rough list of textbook recommendations. There are a ton of Dover paperbacks that I didn't put on here, since they're not as widely used, but they are really great and really cheap.
Amazon search for Dover Books on mathematics
There's also this great list of undergraduate books in math that has become sort of famous: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm
Pre-Calculus / Problem-Solving
Calculus
Linear Algebra
Differential Equations
Number Theory
Proof-Writing
Analysis
Complex Analysis
Functional Analysis
Partial Differential Equations
Higher-dimensional Calculus and Differential Geometry
Abstract Algebra
Geometry
Topology
Set Theory and Logic
Combinatorics / Discrete Math
Graph Theory
P. S., if you Google search any of the topics above, you are likely to find many resources. You can find a lot of lecture notes by searching, say, "real analysis lecture notes filetype:pdf site:.edu"
The rate of your learning is defined by your determination. If you don't give up then you will learn the material.
Look at the book that is required and only learn what you need in the class. Don't learn everything in the book either. Just learn what you need to do well and refer to the books when you get confused.
Note don't try to learn everything that's below. Only use it to learn what you actually need. This can be overwhelming at first but just set aside a set time to study this.
EDIT I added more books and courses.
OCW
http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/
http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/index.htm
http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/
http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/
Helpful books
http://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321390539/ref=sr_1_3?s=books&ie=UTF8&qid=1312542911&sr=1-3
http://www.amazon.com/Understanding-Probability-Chance-Rules-Everyday/dp/0521540364
http://www.amazon.com/gp/product/048663518X/ref=pd_lpo_k2_dp_sr_1?pf_rd_p=486539851&pf_rd_s=lpo-top-stripe-1&pf_rd_t=201&pf_rd_i=0155510053&pf_rd_m=ATVPDKIKX0DER&pf_rd_r=0YXJR9EVHCH9PCBDN372
Khan Academy
http://khan-academy.appspot.com/#calculus
http://www.youtube.com/user/keithpeterb#p/u/19/dS2p_APpcnI
http://khan-academy.appspot.com/video/probability--part-1?playlist=Old%20Algebra
http://www.youtube.com/user/keithpeterb#p/u/19/dS2p_APpcnI
http://khan-academy.appspot.com/video/linear-algebra--introduction-to-vectors?playlist=Linear%20Algebra
EDIT: I knew nothing about topological quantum computation about 1.5 years ago but then I took a independent study in college and I was assigned 1-3 papers a week to read. Eventually I got it a few months ago. What got me through it was not giving up...
OCW
Single Variable Calculus
Multivariable Calculus
Differential Equasions
Linear Algebra
Helpful books
Mathematical Proofs: A Transition to Advanced Mathematics
Understanding Probability: Chance Rules in Everyday Life
Linear Algebra
Other Resources
Kahn Academy
Probability (keithpeterb on youtube)
Probability (Kahn)
Linear Algebra (Kahn)
There, Fixed up your links.
I doubt that you're going to find everything you're looking for in a single book.
I suggest that you start with Axler's Linear Algegra Done Right. Despite the pretentious name it does a good job of introducing linear algebra in a general form.
But Axler doesn't do any applications and almost completely ignores determinants (which I like, but it sounds like you want more of that) so I would supplement with Strang's MIT Lectures and any one of his books.
> Mathematical Logic
It's not exactly Math Logic, just a bunch of techniques mathematicians use. Math Logic is an actual area of study. Similarly, actual Set Theory and Proof Theory are different from the small set of techniques that most mathematicians use.
Also, looks like you have chosen mostly old, but very popular books. While studying out of these books, keep looking for other books. Just because the book was once popular at a school, doesn't mean it is appropriate for your situation. Every year there are new (and quite frankly) pedagogically better books published. Look through them.
Here's how you find newer books. Go to Amazon. In the search field, choose "Books" and enter whatever term that interests you. Say, "mathematical proofs". Amazon will come up with a bunch of books. First, sort by relevance. That will give you an idea of what's currently popular. Check every single one of them. You'll find hidden jewels no one talks about. Then sort by publication date. That way you'll find newer books - some that haven't even been published yet. If you change the search term even slightly Amazon will come up with completely different batch of books. Also, search for books on Springer, Cambridge Press, MIT Press, MAA and the like. They usually house really cool new titles. Here are a couple of upcoming titles that might be of interest to you: An Illustrative Introduction to Modern Analysis by Katzourakis/Varvarouka, Understanding Topology by Shaun Ault. I bet these books will be far more pedagogically sound as compared to the dry-ass, boring compendium of facts like the books by Rudin.
If you want to learn how to do routine proofs, there are about one million titles out there. Also, note books titled Discrete Math are the best for learning how to do proofs. You get to learn techniques that are not covered in, say, How to Prove It by Velleman. My favorites are the books by Susanna Epp, Edward Scheinerman and Ralph Grimaldi. Also, note a lot of intro to proofs books cover much more than the bare minimum of How to Prove It by Velleman. For example, Math Proofs by Chartrand et al has sections about doing Analysis, Group Theory, Topology, Number Theory proofs. A lot of proof books do not cover proofs from Analysis, so lately a glut of new books that cover that area hit the market. For example, Intro to Proof Through Real Analysis by Madden/Aubrey, Analysis Lifesaver by Grinberg(Some of the reviewers are complaining that this book doesn't have enough material which is ridiculous because this book tackles some ugly topological stuff like compactness in the most general way head-on as opposed to most into Real Analysis books that simply shy away from it), Writing Proofs in Analysis by Kane, How to Think About Analysis by Alcock etc.
Here is a list of extremely gentle titles: Discovering Group Theory by Barnard/Neil, A Friendly Introduction to Group Theory by Nash, Abstract Algebra: A Student-Friendly Approach by the Dos Reis, Elementary Number Theory by Koshy, Undergraduate Topology: A Working Textbook by McClusckey/McMaster, Linear Algebra: Step by Step by Singh (This one is every bit as good as Axler, just a bit less pretentious, contains more examples and much more accessible), Analysis: With an Introduction to Proof by Lay, Vector Calculus, Linear Algebra, and Differential Forms by Hubbard & Hubbard, etc
This only scratches the surface of what's out there. For example, there are books dedicated to doing proofs in Computer Science(for example, Fundamental Proof Methods in Computer Science by Arkoudas/Musser, Practical Analysis of Algorithms by Vrajitorou/Knight, Probability and Computing by Mizenmacher/Upfal), Category Theory etc. The point is to keep looking. There's always something better just around the corner. You don't have to confine yourself to books someone(some people) declared the "it" book at some point in time.
Last, but not least, if you are poor, peruse Libgen.
Preach it, brother!
Let me highly recommend Conceptual Mathematics: A First Introduction to Categories and Topoi: The Categorial Analysis of Logic as introductions to the topic requiring no more than a completely rudimentary grasp of set theory to get started--and really, they even motivate the rudimentary set theory. These are basically "Category Theory for Dummies," or at least the closest things that I've found so far.
I would guess that career prospects are a little worse than CS for undergrad degrees, but since my main concern is where a phd in math will take me, you should get a second opinion on that.
Something to keep in mind is that "higher" math (the kind most students start to see around junior level) is in many ways very different from the stuff before. I hated calculus and doing calculations in general, and was pursuing a math minor because I thought it might help with job prospects, but when I got to the more abstract stuff, I loved it. It's easily possible that you'll enjoy both, I'm just pointing out that enjoying one doesn't necessarily imply enjoying the other. It's also worth noting that making the transition is not easy for most of us, and that if you struggle a lot when you first have to focus a lot of time on proving things, it shouldn't be taken as a signal to give up if you enjoy the material.
This wouldn't be necessary, but if you like, here are some books on abstract math topics that are aimed towards beginners you could look into to get a basic idea of what more abstract math is like:
Different mathematicians gravitate towards different subjects, so it's not easy to predict which you would enjoy more. I'm recommending these five because they were personally helpful to me a few years ago and I've read them in full, not because I don't think anyone can suggest better. And of course, you could just jump right into coursework like how most of us start. Best of luck!
(edit: can't count and thought five was four)
if you want determinants, Shilov's is supposed to be "Determinants done right" I wouldn't recommend the other Dover LA book by Stoll
http://www.amazon.com/Linear-Algebra-Dover-Books-Mathematics/product-reviews/048663518X/
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Anyway: Free!
http://www.math.ucdavis.edu/~anne/linear_algebra/
http://www.math.ucdavis.edu/~linear/linear.pdf
http://www.cs.cornell.edu/courses/cs485/2006sp/LinAlg_Complete.pdf (Dawkins notes that were recently pulled off lamar.edu site, gentle intro like Anton's)
http://joshua.smcvt.edu/linearalgebra/
http://www.ee.ucla.edu/~vandenbe/103/reader.pdf
http://www.math.brown.edu/%7Etreil/papers/LADW/LADW.pdf
https://math.byu.edu/~klkuttle/Linearalgebra.pdf
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Or, google "positive definite matrix" or "hermitian" or "hessian" or some term like that and it will show you lecture notes from dozens of universities after the inevitable wikipedia and Wolfram hits
I think the advice given in the rest of the thread is pretty good, though some of it a little naive. The suggestion that differential equations or applied math somehow should not be of interest is silly. A lot of it builds the motivation for some of the abstract stuff which is pretty cool, and a lot of it has very pure problems associated with it. In addition I think after (or rather alongside) your initial calculus education is a good time to look at some other things before moving onto more difficult topics like abstract algebra, topology, analysis etc.
The first course I took in undergrad was a course that introduced logic, writing proofs, as well as basic number theory. The latter was surprisingly useful as it built modular arithmetic which gave us a lot of groups and rings to play with in subsequent algebra courses. Unfortunately the textbook was god awful. I've heard good things about the following two sources and together they seem to cover the content:
How to prove it
Number theory
After this I would take a look at linear algebra. This a field with a large amount of uses in both pure and applied math. It is useful as it will get you used to doing algebraic proofs, it takes a look at some common themes in algebra, matrices (one of the objects studied) are also used thoroughly in physics and applied mathematics and the knowledge is useful for numerical approximations of ordinary and partial differential equations. The book I used Linear Algebra by Friedberg, Insel and Spence, but I've heard there are better.
At this point I think it would be good to move onto Abstract Algebra, Analysis and Topology. I think Farmerje gave a good list.
There's many more topics that you could possibly cover, ODEs and PDEs are very applicable and have a rich theory associated with them, Complex Analysis is a beautiful subject, but I think there's plenty to keep you busy for the time being.
If your goal is mainly to 'understand the adults in the room' then the above is major overkill in my opinion. PCA basically boils down to an application of the Singular Value Decomposition, which is itself a generalization of matrix diagonalization. The book 'Linear Algebra and Its Applications' by David Lay - which is a standard advanced undergraduate text, loaded with examples and great for 'getting the gist' - wraps up with the SVD and a couple of applied examples of using it for PCA.
I'd hazard that you can pretty easily achieve your goals by grasping the SVD and the basic linear algebra concepts that underpin it (multiplication, eigen values, diagonalization and a couple more).
I'll leave you with a site I've had great success with with others for getting to grips with some of the intuition http://www.uwlax.edu/faculty/will/svd/svd/index.html
Please, simply disregard everything below if the info is old news to you.
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Algebraic geometry requires the knowledge of commutative algebra which requires the knowledge of some basic abstract algebra (consists of vector spaces, groups, rings, modules and the whole nine yards). There are many books written on abstract algebra like those of Dummit&Foote, Artin, Herstein, Aluffi, Lang, Jacobson, Hungerford, MacLane/Birkhoff etc. There are a million much more elementary intros out there, though. Some of them are:
Discovering Group Theory: A Transition to Advanced Mathematics by Barnard/Neil
A Friendly Introduction to Group Theory by Nash
Abstract Algebra: A Student-Friendly Approach by the Dos Reis
Numbers and Symmetry: An Introduction to Algebra by Johnston/Richman
Rings and Factorization by Sharpe
Linear Algebra: Step by Step by Singh
As far as DE go, you probably want to see them done rigorously first. I think the books you are looking for are titled something along the lines of "Analysis on Manifolds". There are famous books on the subject by Sternberg, Spivak, Munkres etc. If you don't know basic real analysis, these books will be brutal. Some elementary analysis and topology books are:
Understanding Analysis by Abbot
The Real Analysis Lifesaver by Grinberg
A Course in Real Analysis by Mcdonald/Weiss
Analysis by Its History by Hirer/Wanner
Introductory Topology: Exercises and Solutions by Mortad
I don't say this to be discouraging: Most people don't really have any idea what doing Physics at a high level looks like. I decided in High School that I wanted to be a physicist, and as luck would have it I'm a graduate student and I still enjoy it, but truth be told, the exposure you have in High School doesn't really prepare you for the reality. All that to say: There's no reason to decide at thirteen years old that you need a PhD in Physics! Maybe once you learn math beyond trig you'll decide it isn't for you, or maybe you'll love math and want to switch to a math degree.
All right, now that that's out of the way... You said you're learning trig, that's good, you need it. You also need some basic algebra skills. Then try to teach yourself basic calculus (limits, derivatives, integrals). Then you want to learn Linear Algebra and at least Ordinary Differential Equations.
You can also do some basic physics reading before you've learned the essentials. I really like George Gamow's books for this - he was a very well know and important physicist who also happened to write very accessible books that are very much for lay people but that also don't shy away completely from the math. I really enjoyed this one in particular.
For mathematics, I love Dover books - they're cheap AND good. Shilov, I've found, is clear and readable. This might not be introductory level, but it's inexpensive and let's you see what you're getting yourself into.
Last bit of advice for Physics is what one of my old high school teachers used to say - draw, label, and you can't go wrong. It's still mostly true.
Pick up mathematics. Now if you have never done math past the high school and are an "average person" you probably cringed.
Math (an "actual kind") is nothing like the kind of shit you've seen back in grade school. To break into this incredible world all you need is to know math at the level of, say, 6th grade.
Intro to Math:
These books only serve as samplers because they don't even begin to scratch the surface of math. After you familiarized yourself with the basics of writing proofs you can get started with intro to the largest subsets of math like:
Intro to Abstract Algebra:
There are tons more books on abstract/modern algebra. Just search them on Amazon. Some of the famous, but less accessible ones are
Intro to Real Analysis:
Again, there are tons of more famous and less accessible books on this subject. There are books by Rudin, Royden, Kolmogorov etc.
Ideally, after this you would follow it up with a nice course on rigorous multivariable calculus. Easiest and most approachable and totally doable one at this point is
At this point it's clear there are tons of more famous and less accessible books on this subject :) I won't list them because if you are at this point of math development you can definitely find them yourself :)
From here you can graduate to studying category theory, differential geometry, algebraic geometry, more advanced texts on combinatorics, graph theory, number theory, complex analysis, probability, topology, algorithms, functional analysis etc
Most listed books and more can be found on libgen if you can't afford to buy them. If you are stuck on homework, you'll find help on [MathStackexchange] (https://math.stackexchange.com/questions).
Good luck.
Grad, curl, and div are essentially all the same operation: the exterior derivative.
Grad takes a scalarfield and gives you a vectorfield.
Curl takes a vectorfield and gives you a vectorfield.
Div takes a vectorfield and gives you a scalarfield.
But there's more to it. If you look at the resulting fields from these operations and then perform a change of variables (in manifold speak, you view the same fields in a different chart), they don't look right.
It turns out there is another kind of object called a k-form. These give to every point in space an alternating k-linear form on the tangent space. In other words, you give it k tangent vectors and it will spit out a number. The adjective alternating means if any tangent vector is repeated, the number output is zero. And the number is multilinear (ie: it's linear in each input separately, so doubling the length of any tangent vector doubles the number it spits out).
The 0-forms of a manifold are just scalarfields. The 1-forms are covectorfields. (You give a tangent vector and it spits out a number). As geometric objects, 1-forms look exactly like vectorfields, but they act different under a change of coordinates. You might say they transform "correctly".
A 1-form can be integrated over a curve. The result is a line integral, just like in usual vector calculus. However, because our objects now transform correctly, changing coordinates works as it should.
So grad, rather than being a mapping from scalarfields to vectorfields is actually a mapping from 0-forms (also just scalarfields) to 1-forms (covectorfields).
Similarly, curl isn't a mapping from vectorfields to vetorfields. Instead, it is a mapping from 1-forms to 2-forms.
A 2-form takes two tangent vectors and spits out a number. Intuitively, it returns the signed area spanned by the two tangent vectors.
Note that even though both 1-forms and 2-forms naively sync up with the notion of "vectorfield", they both act differently than a vectorfield and from each other. Under the hood, it has to do with the properties of the wedge product, written ∧, the basic operation for combining forms.
In R^3, we have standard basis x, y, and z. Well, it turns out that the k-forms also have a standard basis. For 1-forms, our basis is dx, dy, and dz. (The covector dx eats a vector and tells you what its x-component was, etc). For 2-forms, we just wedge things together: dx ∧ dy, dy ∧ dz, and dz ∧ dx form a basis. (One of our properties for wedge products is that a ∧ b = -(b ∧ a), so dy ∧ dx wouldn't be included in the basis if dx ∧ dy was). Meanwhile, 3-forms have a basis consisting of just of the triple-wedge dx ∧ dy ∧ dz.
Count the dimension of these spaces. At any point in our manifold, the 1-forms form a 3-dimensional vectorspace and the 2-forms also form a 3-dimenisonal vectorspace... but they have different bases. The 3-forms are 1-dimensional and (trivially) the 0-forms are also 1-dimensional. But again, they are not the same space!
So finally, div is a mapping from 2-forms to 3-forms.
The k-forms are admittedly very convoluted and tricky to work with. They are hardly intuitive compared to what you learn in vector calculus. But they have the advantage of playing nicely with change of coordinates. Maybe more importantly, they also generalize to any number of dimensions. Grad, curl, and div only really work in R^3. But the theory of electromagnetism and the theory of relativity take place in R^4.
For a nice introduction to the subject, you might want to check out Hubbard and Hubbard's excellent and pragmatic introduction to the subject.
Introduction to Linear Algebra is an excellent textbook. Strang explains things in very simple, "what's the point" terms. This is the only textbook I have ever actually enjoyed reading. There are also quite a few videos of Strang's lectures at MIT where he works out plenty of examples.
Schaum's Linear Algebra reads like an exam review: it highlights the main concepts (without the theory) and presents hundreds of worked out examples.
/u/another_user_name posted this list a while back. Actual aerospace textbooks are towards the bottom but you'll need a working knowledge of the prereqs first.
Non-core/Pre-reqs:
Mathematics:
Calculus.
1-4) Calculus, Stewart -- This is a very common book and I felt it was ok, but there's mixed opinions about it. Try to get a cheap, used copy.
1-4) Calculus, A New Horizon, Anton -- This is highly valued by many people, but I haven't read it.
1-4) Essential Calculus With Applications, Silverman -- Dover book.
More discussion in this reddit thread.
Linear Algebra
3) Linear Algebra and Its Applications,Lay -- I had this one in school. I think it was decent.
3) Linear Algebra, Shilov -- Dover book.
Differential Equations
4) An Introduction to Ordinary Differential Equations, Coddington -- Dover book, highly reviewed on Amazon.
G) Partial Differential Equations, Evans
G) Partial Differential Equations For Scientists and Engineers, Farlow
More discussion here.
Numerical Analysis
5) Numerical Analysis, Burden and Faires
Chemistry:
Physics:
2-4) Physics, Cutnel -- This was highly recommended, but I've not read it.
Programming:
Introductory Programming
Programming is becoming unavoidable as an engineering skill. I think Python is a strong introductory language that's got a lot of uses in industry.
Core Curriculum:
Introduction:
Aerodynamics:
Thermodynamics, Heat transfer and Propulsion:
Flight Mechanics, Stability and Control
5+) Flight Stability and Automatic Control, Nelson
5+)[Performance, Stability, Dynamics, and Control of Airplanes, Second Edition](http://www.amazon.com/Performance-Stability-Dynamics-Airplanes-Education/dp/1563475839/ref=sr_1_1?ie=UTF8&qid=1315534435&sr=8-1, Pamadi) -- I gather this is better than Nelson
Engineering Mechanics and Structures:
3-4) Engineering Mechanics: Statics and Dynamics, Hibbeler
6-8) Analysis and Design of Flight Vehicle Structures, Bruhn -- A good reference, never really used it as a text.
G) Introduction to the Mechanics of a Continuous Medium, Malvern
G) Fracture Mechanics, Anderson
G) Mechanics of Composite Materials, Jones
Electrical Engineering
Design and Optimization
Space Systems
There are essentially "two types" of math: that for mathematicians and everyone else. When you see the sequence Calculus(1, 2, 3) -> Linear Algebra -> DiffEq (in that order) thrown around, you can be sure they are talking about non-rigorous, non-proof based kind that's good for nothing, imo of course. Calculus in this sequence is Analysis with all its important bits chopped off, so that everyone not into math can get that outta way quick and concentrate on where their passion lies. The same goes for Linear Algebra. LA in the sequence above is absolutely butchered so that non-math majors can pass and move on. Besides, you don't take LA or Calculus or other math subjects just once as a math major and move on: you take a rigorous/proof-based intro as an undergrad, then more advanced kind as a grad student etc.
To illustrate my point:
Linear Algebra:
Linear Algebra Through Geometry by Banchoff and Wermer
3. Here's more rigorous/abstract Linear Algebra for undergrads:
Linear Algebra Done Right by Axler
4. Here's more advanced grad level Linear Algebra:
Advanced Linear Algebra by Steven Roman
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Calculus:
Calulus by Spivak
3. Full-blown undergrad level Analysis(proof-based):
Analysis by Rudin
4. More advanced Calculus for advance undergrads and grad students:
Advanced Calculus by Sternberg and Loomis
The same holds true for just about any subject in math. Btw, I am not saying you should study these books. The point and truth is you can start learning math right now, right this moment instead of reading lame and useless books designed to extract money out of students. Besides, there are so many more math subjects that are so much more interesting than the tired old Calculus: combinatorics, number theory, probability etc. Each of those have intros you can get started with right this moment.
Here's how you start studying real math NOW:
Learning to Reason: An Introduction to Logic, Sets, and Relations by Rodgers. Essentially, this book is about the language that you need to be able to understand mathematicians, read and write proofs. It's not terribly comprehensive, but the amount of info it packs beats the usual first two years of math undergrad 1000x over. Books like this should be taught in high school. For alternatives, look into
Discrete Math by Susanna Epp
How To prove It by Velleman
Intro To Category Theory by Lawvere and Schnauel
There are TONS great, quality books out there, you just need to get yourself a liitle familiar with what real math looks like, so that you can explore further on your own instead of reading garbage and never getting even one step closer to mathematics.
If you want to consolidate your knowledge you get from books like those of Rodgers and Velleman and take it many, many steps further:
Basic Language of Math by Schaffer. It's a much more advanced book than those listed above, but contains all the basic tools of math you'll need.
I'd like to say soooooooooo much more, but I am sue you're bored by now, so I'll stop here.
Good Luck, buddyroo.
First, Sheldon Axler's Linear Algebra Done Right is a favorite of many folks. It might be a little much as supplementary material, but it will likely give you a very fresh perspective on whatever material you're covering in class.
Second, disabuse yourself of the idea that "vectors" are little arrows sitting in some n-dimensional space. Vectors are not the important concept, vector spaces are. A vector is just a name for an element of a vector space and they may or may not be representable as a little arrow or a finite n-tuple (x1, x2, ..., xn). A vector space over the real numbers is any collection of objects which satisfy a particular set of properties.
For example, the space of all continuous functions on the real line f: ℝ → ℝ, defined as
> C^(0)(ℝ) = {f: ℝ → ℝ | f is continuous}
is a vector space that doesn't look anything like ℝ^(n) (n-dimensional Euclidean space). We can talk about maps between vector spaces which preserve the vector space structure, which we call linear maps or linear transformations. Matrices are one way of representing linear maps and matrix "multiplication" is defined so that the product of two matrices corresponds to the composition of the linear maps they represent.
That fact is probably the first "ah ha!" moment for a lot of students that makes them realize there's more going on in linear algebra than just a bunch of vector/matrix manipulation.
In your first linear algebra course you'll likely be focusing on ℝ^(n), which has a bunch of extra structure relative to a plain vector space. It's very easy for a first-time linear algebra student to conflate all the extra "stuff" that comes with the structure of ℝ^(n) with the totality of linear algebra. So keep an eye out!
Third, you might be tempted to relate what you've done with matrices in the context of game programming to what you're learning in your linear algebra class as quickly as possible. Be careful because the way computers use matrices is slightly more complex than the material you'll learn early on. Specifically, folks tend to use homogeneous coordinates when writing games, which is why libraries like OpenGL use 4×4 matrices to represent transformations of 3d space instead of 3×3 matrices (and, similarly, 3×3 matrices to represent transformations of 2d space instead of 2×2 matrices as you might naïvely expect).
We do this because computers (GPUs, really) are built to manipulate matrices very, very quickly, so we ♥ it when the transforms we want to perform can be express as a matrix. Unfortunately, translations in n-dimensional space can't be represented as n×n matrices. However, if we move to real projective space then there is a way to represent translations as matrices in the projective space. But points in n-dimensional projective space can be represented by n+1 coordinates, so while there is no n×n matrix representing a translations in n-dimensional space there are corresponding (n+1)×(n+1) matrices which represent that translation in n-dimensional projective space.
I found Axler's Linear Algebra Done Right to be a very easy to digest introduction to abstract linear algebra.
If you are serious about learning, Linear Algebra by Friedberg Insel and Spence, or Linear Algebra by Greub are your best bets. I love both books, but the first one is a bit easier to read.
You're not really doing higher math right now as much as you're learning tricks to solve problems. Once you start proving stuff that'll be a big jump. Usually people start doing that around Real Analysis like your father said. Higher math classes almost entirely consist of proofs. It's a lot of fun once you get the hang of it, but if you've never done it much before it can be jarring to learn how. The goal is to develop mathematical maturity.
Start learning some geometry proofs or pick up a book called "Calculus" by Spivak if you want to start proving stuff now. The Spivak book will give you a massive head start if you read it before college. Differential equations will feel like a joke after this book. It's called calculus but it's really more like real analysis for beginners with a lot of the harder stuff cut out. If you can get through the first 8 chapters or so, which are the hardest ones, you'll understand a lot of mathematics much more deeply than you do now. I'd also look into a book called Linear Algebra done right. This one might be harder to jump into at first but it's overall easier than the other book.
This book has good Google reviews. I haven't read it.
If you know topology and algebra then I think the most fruitful way of approaching categories is by picking up a book on algebraic topology. Hatcher is a canonical reference and although it doesn't really introduce the formal language of category theory, it is shown through most of the book. If you're not that patient and have more mathematical maturity then just pick up May's concise course in algebraic topology which is a wild ride but will get you there.
The most canonical textbook is Mac Lane's categories for the working mathematician but it's kind of dry so you'll need to provide your own class of examples every time.
If you just want to take a look into the topic and see how it is like then I would recommend you to read the first three chapters of this book. The main topic of the book is a little bit advanced so you can ignore any mention of topoi but those first three capters are a very brief introduction to category theory through a couple of examples and as far as I remember it doesn't expect you to know the stuff beforehand. It is however very basic and it won't cover a lot of the useful constructions insipired in algebraic topology/geometry but I still believe it's a pretty nice summary for the language.
Many people like Strang's book for a standard treatment. Then getting a bit more general is Axler's Linear Algebra Done Right. But for further understanding, especially for computer graphics I recommend learning about the exterior and geometric algebras.
https://www.amazon.com/Linear-Algebra-via-Exterior-Products/dp/140929496X/
https://www.amazon.com/Linear-Geometric-Algebra-Alan-Macdonald/dp/1453854932
I think the most important part of being able to see beauty in mathematics is transitioning to texts which are based on proofs rather than application. A side effect of gaining the ability to read and write proofs is that you're forced to deeply understand the theory of the math you're learning, as well as actively using your intuition to solve problems, rather than dry route calculations found in most application based textbooks. Based on what you've written, I feel you may enjoy taking this path.
Along these lines, you could start of with Book of Proof (free) or How to Prove It. From there, I would recommend starting off with a lighter proof based text, like Calculus by Spivak, Linear Algebra Done Right by Axler, or Pinter's book as you mentioned. Doing any intro proofs book plus another book at the level I mentioned here would have you well prepared to read any standard book at the undergraduate level (Analysis, Algebra, Topology, etc).
You need some grounding in foundational topics like Propositional Logic, Proofs, Sets and Functions for higher math. If you've seen some of that in your Discrete Math class, you can jump straight into Abstract Algebra, Rigorous Linear Algebra (if you know some LA) and even Real Analysis. If thats not the case, the most expository and clearly written book on the above topics I have ever seen is Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers.
Some user friendly books on Real Analysis:
Some user friendly books on Linear/Abstract Algebra:
Topology(even high school students can manage the first two titles):
Some transitional books:
Plus many more- just scour your local library and the internet.
Good Luck, Dude/Dudette.
I haven't used it myself, but you might appreciate Gilbert Strang's Linear Algebra and Its Applications.
I think it is silly that this requires much of an explanation. I recall this question being asked in a fucking linear algebra book! http://www.amazon.com/Linear-Algebra-Applications-Gilbert-Strang/dp/0030105676/ref=dp_ob_title_bk
Linear Algebra Done Right is a good introduction, but if you want to go beyond an undergraduate level, try Linear Algebra by Hoffman and Kunze.
Intro Calculus, in American sense, could as well be renamed "Physics 101" or some such since it's not a very mathematical course. Since Intro Calculus won't teach you how to think you're gonna need a book like How to Solve Word Problems in Calculus by Eugene Don and Benay Don pretty soon.
Aside from that, try these:
Excursions In Calculus by Robert Young.
Calculus:A Liberal Art by William McGowen Priestley.
Calculus for the Ambitious by T. W. KORNER.
Calculus: Concepts and Methods by Ken Binmore and Joan Davies
You can also start with "Calculus proper" = Analysis. The Bible of not-quite-analysis is:
[Calculus by Michael Spivak] (http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?s=books&ie=UTF8&qid=1413311074&sr=1-1&keywords=spivak+calculus).
Also, Analysis is all about inequalities as opposed to Algebra(identities), so you want to be familiar with them:
Introduction to Inequalities by Edwin F. Beckenbach, R. Bellman.
Analytic Inequalities by Nicholas D. Kazarinoff.
As for Linear Algebra, this subject is all over the place. There is about a million books of all levels written every year on this subject, many of which is trash.
My plan would go like this:
1. Learn the geometry of LA and how to prove things in LA:
Linear Algebra Through Geometry by Thomas Banchoff and John Wermer.
Linear Algebra, Third Edition: Algorithms, Applications, and Techniques
by Richard Bronson and Gabriel B. Costa.
2. Getting a bit more sophisticated:
Linear Algebra Done Right by Sheldon Axler.
Linear Algebra: An Introduction to Abstract Mathematics by Robert J. Valenza.
Linear Algebra Done Wrong by Sergei Treil.
3. Turn into the LinAl's 1% :)
Advanced Linear Algebra by Steven Roman.
Good Luck.
Hrrumph. Determinants are a capstone, not a cornerstone, of Linear Algebra.
https://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mathematics/dp/0387982582
>When university starts, what can I do to ensure that I can compete and am just as good as the best mathematics students?
Read textbooks for mathematics students.
For example for Linear Algebra I heard that Axler's book is very good (I studied Linear Algebra in another language, so I can't really suggest anything from personal experience). For Calculus I personally suggest Spivak's book.
There are many books that I could suggest, but one of the greatest books I've ever read is The Art and Craft of Problem Solving.
Category theory is not easy to get into, and you have to learn quite a bit and use it for stuff in order to retain a decent understanding.
The best book for an introduction I have read is:
Algebra (http://www.amazon.com/Algebra-Chelsea-Publishing-Saunders-Lane/dp/0821816462/ref=sr_1_1?ie=UTF8&qid=1453926037&sr=8-1&keywords=algebra+maclane)
For more advanced stuff, and to secure the understanding better I recommend this book:
Topoi - The Categorical Analysis of Logic (http://www.amazon.com/Topoi-Categorial-Analysis-Logic-Mathematics/dp/0486450260/ref=sr_1_1?ie=UTF8&qid=1453926180&sr=8-1&keywords=topoi)
Both of these books build up from the basics, but a basic understanding of set theory, category theory, and logic is recommended for the second book.
For type theory and lambda calculus I have found the following book to be the best:
Type Theory and Formal Proof - An Introduction (http://www.amazon.com/Type-Theory-Formal-Proof-Introduction/dp/110703650X/ref=sr_1_2?ie=UTF8&qid=1453926270&sr=8-2&keywords=type+theory)
The first half of the book goes over lambda calculus, the fundamentals of type theory and the lambda cube. This is a great introduction because it doesn't go deep into proofs or implementation details.
Have a look at Schaum's outline of Linear Algebra. Pdf
I've never seen a math textbook as incredible and amazing as that one.
The way determinants are presented (permutations + multilinear algebra!!!!!) is absolutely incredible.
The material on group theory, direct sums, tensor products, etc... is presented at a very accessible level.
Tensors are introduced as elements of tensor spaces rather than introducing tensors as "objects that rotate like tensors" (am a physics student).
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Absolutely recommend you buy the amazon copy (14$ - it's extremely cheap!!).
https://www.amazon.com/Schaums-Outline-Linear-Algebra-5th/dp/0071794565
David Lay's Linear Algebra and Its Applications
Lecture Notes I used (Go to middle of page)
Linear algebra is an essential tool in many areas of mathematics. Computations with matrices aren't always that important; far more important are the concepts of vector space and linear transformation. Pretty much any time you work with coordinates, dimension, changes of coordinates, vectors, linear relations, or anything like that, you're going to need some linear algebra.
If you're interested, I recommend taking a look at Axler's Linear Algebra Done Right. Axler has very clear exposition and proofs, and if you've only seen the computational aspect of linear algebra, it'll provide a different, more abstract and conceptual perspective.
For anyone else, I assume the specific book is Goldblatt's "Topoi: The Categorical Analysis of Logic"
Here is a link to John Baez's overview of what Topos theory is.
As /u/ziggurism has already said, you don't need to understand algebraic geometry to understand the theory of elementary toposes but some of the early motivating examples, the category of sheaves on a grothendieck site, are heavily steeped in the language of modern algebraic geometry.
A good, non algebro-geometric introduction to topos theory for those seeking to understand its place in logic is the book 'Topoi: The Categorial Analysis of Logic by Goldblatt. It also serves as a great introduction to the ideas of category theory.
Another fantastic book is Lawvere/Roserugh's Sets for Mathematics. This book seeks to explain the axioms of set theory using the language of category theory. It's not a book on arbitrary topos but it does serve to give you an idea of how topos theory axioms serve to build the logical system that every mathematician is familiar with, the logic in the category of sets. It's a good idea to have this 'concrete' application of topos axioms in Set under your belt before you tackling a book that seeks to explain how an arbitrary topos gives you a more abstract and unfamiliar logical system.
Edit: Also worth looking into is how topos theory can be used in the foundations of physics
These were the most enlightening for me on their subjects:
This should keep you busy, but I can suggest books in other areas if you want.
Math books:
Algebra: http://www.amazon.com/Algebra-I-M-Gelfand/dp/0817636773/ref=sr_1_1?ie=UTF8&s=books&qid=1251516690&sr=8
Calc: http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?s=books&ie=UTF8&qid=1356152827&sr=1-1&keywords=spivak+calculus
Calc: http://www.amazon.com/Linear-Algebra-Dover-Books-Mathematics/dp/048663518X
Linear algebra: http://www.amazon.com/Linear-Algebra-Modern-Introduction-CD-ROM/dp/0534998453/ref=sr_1_4?ie=UTF8&s=books&qid=1255703167&sr=8-4
Linear algebra: http://www.amazon.com/Linear-Algebra-Dover-Mathematics-ebook/dp/B00A73IXRC/ref=zg_bs_158739011_2
Beginning physics:
http://www.amazon.com/Feynman-Lectures-Physics-boxed-set/dp/0465023827
Advanced stuff, if you make it through the beginning books:
E&M: http://www.amazon.com/Introduction-Electrodynamics-Edition-David-Griffiths/dp/0321856562/ref=sr_1_1?ie=UTF8&qid=1375653392&sr=8-1&keywords=griffiths+electrodynamics
Mechanics: http://www.amazon.com/Classical-Dynamics-Particles-Systems-Thornton/dp/0534408966/ref=sr_1_1?ie=UTF8&qid=1375653415&sr=8-1&keywords=marion+thornton
Quantum: http://www.amazon.com/Principles-Quantum-Mechanics-2nd-Edition/dp/0306447908/ref=sr_1_1?ie=UTF8&qid=1375653438&sr=8-1&keywords=shankar
Cosmology -- these are both low level and low math, and you can probably handle them now:
http://www.amazon.com/Spacetime-Physics-Edwin-F-Taylor/dp/0716723271
http://www.amazon.com/The-First-Three-Minutes-Universe/dp/0465024378/ref=sr_1_1?ie=UTF8&qid=1356155850&sr=8-1&keywords=the+first+three+minutes
Learn math first. Physics is essentially applied math with experiments. Start with Calculus then Linear Algebra then Real Analysis then Complex Analysis then Ordinary Differential Equations then Partial Differential Equations then Functional Analysis. Also, if you want to pursue high energy physics and/or cosmology, Differential Geometry is also essential. Make sure you do (almost) all the exercises in every chapter. Don't just skim and memorize.
This is a lot of math to learn, but if you are determined enough you can probably master Calculus to Real Analysis, and that will give you a big head start and a deeper understanding of university-level physics.
Congrats! And sorry about the DNF.
My opinion (for whatever its worth i guess), if your right on the edge of cut off times then you have to look at 3 things: age, weight, time spent training.
Unfortunately not much we can do about age, at a certain point no one is finishing a half ironman. I assume that you are not at that age yet.
Weight is probably the hardest thing to adjust. You can't out run a bad diet. So knowing nothing about your weight, are you satisfied with your weight or do you think that there is room for improvement?
Time spent training is the easy stuff! Woooo! More specifically, effective training and an effective training plan is probably your biggest gap. I (and others) suggest a book called The Triathlete's Training Bible by Joel Friel. This gets into how to spend your time to be more effectively training with self guided training plans etc etc. If you give more information about what you did to train for this specific event then maybe we could have more in-depth conversation about what you should be doing.
https://www.amazon.com/Triathletes-Training-Bible-Joe-Friel/dp/1934030198/ref=sr_1_2?ie=UTF8&qid=1491248736&sr=8-2&keywords=triathletes+training+bible
I used Joe Friel's Going Long: Training for Triathlon's Ultimate Challenge and Joe Friel's The Triathlete's training Bible Very in-depth books on how to set up a training plan and schedule your time.
Yes!
If you don't know any calculus Stewart Calculus is the typical primer in colleges. Combine this with Khan Academy for easy mode cruise control.
After that, you want to look at The Big Orange Book, which is essentially the bible for undergrad astrophysics and 100% useful beyond that. This book could, alone, tell you everything you need to know.
As for other topics like differential equations and linear algebra you can shop around. I liked Linear Algebra Done Right for linear personally. No recommendations from me on differential equations though, never found a book that I loved.
I recommend reading the Triathlete's Training Bible (http://www.amazon.com/The-Triathletes-Training-Bible-Friel/dp/1934030198) which quite extensively covers the base training period.
If I recall correctly, he speaks about doing lots of leg and core strength training, swimming drills concentrating heavily on technique, hill repeats on the treadmill, etc... Things that would serve as a good base for other training later on.
Here's a couple of books I'd recommend.
amazon
amazon
You might check out the Minneapolis area for a tri club. I'm certain there is a good one up there. Some clubs have New Triathlete programs that can be really good.
The course I took as an undergraduate used Friedberg, Insel and Spence. I remember liking it fine, but it's insultingly expensive. Find it in a library or get a used copy if you can. If you're looking for a bargain, it can't hurt to try Shilov. He's Russian, so the book is very terse, but covers a lot of ground.
This is a compilation of what I gathered from reading on the internet about self-learning higher maths, I haven't come close to reading all this books or watching all this lectures, still I hope it helps you.
General Stuff:
The books here deal with large parts of mathematics and are good to guide you through it all, but I recommend supplementing them with other books.
Linear Algebra: An extremelly versatile branch of Mathematics that can be applied to almost anything, also the first "real math" class in most universities.
Calculus: The first mathematics course in most Colleges, deals with how functions change and has many applications, besides it's a doorway to Analysis.
Real Analysis: More formalized calculus and math in general, one of the building blocks of modern mathematics.
Abstract Algebra: One of the most important, and in my opinion fun, subjects in mathematics. Deals with algebraic structures, which are roughly sets with operations and properties of this operations.
There are many other beautiful fields in math full of online resources, like Number Theory and Combinatorics, that I would like to put recommendations here, but it is quite late where I live and I learned those in weirder ways (through olympiad classes and problems), so I don't think I can help you with them, still you should do some research on this sub to get good recommendations on this topics and use the General books as guides.
The only possible issue I see is your selection of textbook: Principles of Mathematical Economics - I've honestly never heard of this book.
The graduate school go-to textbook is Mathematics for Economists by Simon and Blume. However, I think this book would be overkill for you, as it is geared towards pure, PhD level, economics. Also, I was in a similar place to you, with respect to mathematical training at one point, and Simon & Blume proved to be too large a leap.
My advice would be to use one of the following books (in order of my preference):
1. Essential Mathematics for Economic Analysis by Sydsaeter
2. Mathematics for Economics
by Hoy
3. Fundamental Methods of Mathematical Economics
by Chiang
They'll bring your basic command, of the basic required mathematics up to scratch AND these books cover linear algebra. You will also then be in a good place to tackle Simon & Blume if you ever need to in the future. Another piece of advice: PRACTISE PRACTISE PRACTISE. For what you are doing, you don't need to have a deep understanding of the mathematics you are using BUT, you do need to be very comfortable with applying the techniques.
So, as you are working through (for instance) Sydsaeter, I would be attempting the related practice questions you find in:
Hope this helps.
P.S. Almost all of these books are available for 'free' on Library Gensis
A First Course in Graph Theory by Chartrand and Zhang
Combinatorics: A Guided Tour by Mazur
Discrete Math by Epp
For Linear Algebra I like these below:
Lecture Notes by Tao
Linear Algebra: An Introduction to Abstract Mathematics by Robert Valenza
Linear Algebra Done Right by Axler
Linear Algebra by Friedberg, Insel and Spence
A graph theory project! I just started today (it was assigned on Friday and this is when I selected my topic). I’m on spring break but next month I have to present a 15-20 minute lecture on graph automorphisms. I don’t necessarily have to, but I want to try and tie it in with some group theory since there is a mix of undergrads who the majority of them have seen some algebra before and probably bored PhD students/algebraists in my class, but I’m not sure where to start. Like, what would the binary operation be, composition of functions? What about the identity and inverse elements, what would those look like? In general, what would the elements of this group look like? What would the group isomorphism be? That means it’s a homomorphism with a bijective function. What would the homomorphism and bijective function look like? These are the questions I’m trying to get answers to.
Last semester I took a first course in Abstract Algebra and I’m currently taking a follow up course in Linear Algebra (I have the same professor for both algebra classes and my graph theory class). I’m curious if I can somehow also bring up some matrix representation theory stuff as that’s what we’re going over in my linear algebra class right now.
This is the textbook I’m using for my graph theory class: Graph Theory (Graduate Texts in Mathematics) https://www.amazon.com/dp/1846289696?ref=yo_pop_ma_swf
Here are the other graph theory books I got from my library and am using as references: Graph Theory (Graduate Texts in Mathematics) https://www.amazon.com/dp/3662536218?ref=yo_pop_ma_swf
Modern Graph Theory (Graduate Texts in Mathematics) https://www.amazon.com/dp/0387984887?ref=yo_pop_ma_swf
And for funsies, here is my linear algebra text: Linear Algebra, 4th Edition https://www.amazon.com/dp/0130084514?ref=yo_pop_ma_swf
But that’s what I’m working on! :)
And I certainly wouldn’t mind some pointers or ideas or things to investigate for this project! Like I said, I just started today (about 45 minutes ago) and am just trying to get some basic questions answered. From my preliminary investigating in my textbook, it seems a good example to work with in regards to a graph automorphism would be the Peterson Graph.
Barbara Burke Hubbard, John H. Hubbard:
Vector Calculus, Linear Algebra and Differential Forms A Unified Approach (1998)
Contents: http://i.imgur.com/1Hj4h52.png
Amazon
Other people may recommend other versions but the important part is the author Hubbard.
Linear Algebra and [Linear Algebra and Its Applications] (http://www.amazon.com/dp/0321385179).
We only need Pythagorean theorem to understand special relativity. Consider two dudes X and Y. Suppose X is on a flying carpet holding up two mirrors distance of h apart. Also assume there's a light particle bouncing between these mirrors vertically like this here. So we see that h = ct_x where c is the speed of light and t_x is the amount of time it takes for the light to go from one mirror to the other. Now have Y stand on the ground and observe the behavior of the light particle as the carpet flies horizontally. From the perspective of Y, the particle flies in a sawtooth pattern like this. The distance the particle travels diagonally depends on the speed of light c and so it is ct_y where t_y is time taken by light to bounce from one mirror to the other as seen by Y. The distance the particle travels horizontally depends on the speed s of carpet and so it is st_y. By Pythagorean theorem, we have h^2 + (st_y)^2 = (ct_y)^2 which implies (t_y)^2 = h^2 / (c^2 (1 - s^2 / c^2 )) which further implies t_y = t_x / (sqrt(1 - s^2 / c^2 )). Thus if s = 0, then t_x = t_y and so time is universal. But as s approaches the speed of light c, the clocks desynchronize.
@ OP, if you want to get into high-falutin physics, you want to know the basics of real, functional (covers linear algebra), complex analyses; some probability and statistics; a bit of group theory.
For analysis the books by Lara Alcock, Amol Sasane, Paul Zorn, Robert Strichartz, Jonathan Kane, Steven Lay, Stephen Abbot, K.G Binmore, Charles Pugh, Mary Hart and many others are very user-friendly. And taking into account your background, Linear Algebra: Step by Step by Kuldeep Singh is perfect for you.
Hey! I am a math major at Harvey Mudd College (who went to high school in the Pacific NW!). I'll answer from what I've seen.
End: Also, if you wanna learn something cool, I'd check out Discrete math. It's usually required for both a math or CS major, and it's some of the coolest undergraduate math out there. Oh, and, unlike some other math, it's not terrible to self-teach. :)
Good luck! Math is awesome!
Reading a bunch about Turbulence and Topoi, mostly Tsinober and Goldblatt. Working on a proof about the relation of two manifolds also, basicly a lot of Jacobians.
My favorite linear algebra text is Paul Halmos' Finite-Dimensional Vector Spaces. As far as textbooks go, it's cheap, and it's written very well. It does expect a certain amount of mathematical maturity (a familiarity with proof techniques).
Gilbert Strang's book, Linear Algebra and Its Applications might be better for someone looking into applied mathematics than Halmos'. He makes frequent references to applications and uses geometric arguments fairly liberally. It is 3 times the price of Halmos' text as well, but I'm sure your university library has a copy or two.
I agree with urish, that learning linear algebra fairly well, especially considering the fields that you're interested in.
Hope this helps.
You'll usually find the following recommended:
I've personally used Friedberg's text, and I found it to be pretty well written.
I am not sure that a pure math textbook is what you want. A lot of the problems that mathematicians think about may not be what you need. Let's take functional analysis for example. Most textbooks focus on bounded/ compact operators, and they only have one chapter at the end dedicated to unbounded operators. Unfortunately, the derivative (momentum) is an unbounded operator, so the part that has the least detail is what you need.
I would recommend a "math for physics students" book. A nice book that tries to paint the intuitive idea of most branches of math relevant to physics (and then some) and show you how to calculate is Goldbart and Stone's book, which they have made freely available online. This book assumes familiarity with linear algebra. If you are weak on this subject, I would highly recommend the book by Friedberg, Insel, and Spence. This is a more traditional math textbook, but it gets you very comfortable with the details of linear algebra (except for tensor products, but you should understand their construction with this background).
Some possibilities:
Calc I & II: Spivak's Calculus
Calc III and a bit of linear algebra: Hubbard & Hubbard's Vector Calculus
LA: Axler or Shilov or both
ODE: Morris Tanenbaum
Discrete/Combinatorics/etc.: Knuth's Concrete Mathematics
For book suggestions beyond concerning Analysis, Algebra, and Topology, the search box will turn up a ton of previous conversations.
https://www.amazon.com/Vector-Calculus-Linear-Algebra-Differential/dp/0136574467
Plus you get to learn differential forms!
Have you seen Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach by Hubbard&Hubbard ?
I think Linear Algebra by Kuldeep Singh is the best fit for newcomers to LA. It's unpretentious and meant to be actually read by students (can you imagine?). This book will take you from someone who just discovered there exists such a thing as LA to someone who solves problems in Linear Algebra Done Right By Axler cold. After Kuldeep Singh you can pick up Advanced Linear Algebra by Steven Roman which is an extreme overkill even for mathematicians.
Basically, once you get the basics of LA down, you can simply read up on the newest matrix algos for machine learning on ArXiv or something. BTW, if your goal is working with data you need to learn some probability.
Linear Algebra: Step by Step https://www.amazon.com/dp/0199654441/ref=cm_sw_r_cp_api_6l2QzbNAXRANT
This is what I used when I was learning it on my own. It explains things very well and has lots of practice with the correct and worked out answers online.
Lay's Linear Algebra is a great introduction to linear algebra book.
For the intuition behind linear algebra, watch Essence of linear algebra.
Perhaps you might find Shilov's Linear Algebra or Roman's Advanced Linear Algebra to be useful. Both of them treat bilinear and quadratic forms.
I think Shilov does actually discuss Gram-Schmidt orthonormalization, but he doesn't call it that, and it seems to be spread over several sections in chapters 7 and 8. Roman might be better for that. Anyway, you can peruse both of these at libgen.
If you are still an undergrad and your school offers a "how to prove stuff and how to think about abstract maths" course take it anyway. No matter how far along you have come.
An example text for such a course is this one:
https://www.amazon.com/Introduction-Mathematical-Reasoning-Numbers-Functions/dp/0521597188
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As for Linear Algebra (the most useful part of all higher mathematics for sure (R/math: if you disagree, fight me on this one...i'll win) ) I will tell you i learned a LOT from these two texts:
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https://www.amazon.com/Linear-Algebra-Introduction-Mathematics-Undergraduate/dp/0387940995
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https://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mathematics/dp/3319110799/ref=pd_lpo_sbs_14_img_0?_encoding=UTF8&psc=1&refRID=APH3PQE76V9YXKWWGCR9
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If you're dislike of linear algebra comes from using the determinant and matrix calculations, you would love Axler's Linear Algebra Done Right.
I learned a lot from getting a copy of Rudin (however, this book is very challenging and probably not the best to self study from. I was able to get to about continuity before taking my analysis course and it was challenging, but worth while). You can probably find it online somewhere for free.
A teacher lent Introduction to Analysis to me and suggested I use it instead of the book by Rudin. It was a well written book and had exercises which were much more approachable (although it included very difficult ones as well). The layout of this book (and I'd bet many others) is quite similar to that of Rudin. It was nice to be able to read them together.
For linear algebra, I can't speak to the quality of many books, but there are plenty which can fairly easily be found online. You will likely be recommended Linear Algebra Done Right however I found it a bit challenging as a first introduction to linear algebra and never got quite far.
My university course used Larson, Falvo Linear Algebra and it was enjoyable and helps you learn the computations very well and gives a decent understanding of proofs.
There are some really good books that you can use to give yourself a solid foundation for further self-study in mathematics. I've used them myself. The great thing about this type of book is that you can just do the exercises from one side of the book to the other and then be confident in the knowledge that you understand the material. It's nice! Here are my recommendations:
First off, three books on the basics of algebra, trigonometry, and functions and graphs. They're all by a guy called Israel Gelfand, and they're good: Algebra, Trigonometry, and Functions and Graphs.
Next, one of two books (they occupy the same niche, material-wise) on general proof and problem-solving methods. These get you in the headspace of constructing proofs, which is really good. As someone with a bachelors in math, it's disheartening to see that proofs are misunderstood and often disliked by students. The whole point of learning and understanding proofs (and reproducing them yourself) is so that you gain an understanding of the why of the problem under consideration, not just the how... Anyways, I'm rambling! Here they are: How To Prove It: A Structured Approach and How To Solve It.
And finally a book which is a little bit more terse than the others, but which serves to reinforce the key concepts: Basic Mathematics.
After that you have the basics needed to take on any math textbook you like really - beginning from the foundational subjects and working your way upwards, of course. For example, if you wanted to improve your linear algebra skills (e.g. suppose you wanted to learn a bit of machine learning) you could just study a textbook like Linear Algebra Done Right.
The hard part about this method is that it takes a lot of practice to get used to learning from a book. But that's also the upside of it because whenever you're studying it, you're really studying it. It's a pretty straightforward process (bar the moments of frustration, of course).
If you have any other questions about learning math, shoot me a PM. :)
I learned lin. alg. from Axler's Linear Algebra Done Right. I found it extremely readable, with exercises that were not too hard to get through quickly.
I'd suggest Probability, Linear Algebra, Convex Optimization and ML in that order.
As for study materials, I'd suggest
That should keep you busy for a while.
If you'd like an alternative to calculus, try learning linear and/or abstract algebra. Shilov's Linear Algebra is a good book on linear algebra. Linear algebra comes up everywhere, so it's definitely worth learning. The abstractions involved such as fields should also be a good introduction to higher mathematics. For even more abstraction, try A Book of Abstract Algebra by Charles Pinter which is one of my favorite books.
While calculus is also fundamental, personally I find linear and abstract algebra to be much more enjoyable subjects.
Shilov gives a rigorous, determinant-heavy treatment of LA in his $10 book. All the nice properties of determinants are verified in the first chapter
This Book Is a great read. Explains every part of training and competing at your best.
You should also pick up a copy of The Triathlete's Training Bible. It's a great read with lots of good training & nutrition advice.
The Joe Friel Books are great. The Triathletes Training Bible by Joe Friel is fantastic (https://www.amazon.com/Triathletes-Training-Bible-Joe-Friel/dp/1934030198) in addition I found a subscription to training peaks with a training plan to be great for accountability.
Book of proof is a more gentle introduction to proofs then How to Prove it.
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No bullshit guide to linear algebra is a gentle introduction to linear algebra when compared to the popular Linear Algebra Done Right.
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An Illustrated Theory of Numbers is a fantastic introduction book to number theory in a similar style to the popular Visual Complex Analysis.
Where I teach they use Linear Algebra by Lay for the introductory class. I'm not sure what level you need but Linear Algebra Done Right is also commonly recommended; could be more abstract than what you need?
A matrix "times" a vector is a vector. This is how the matrix behaves as a linear transformation. (Hence why everyone is telling you you're talking about linear transformations.) To get a matrix to behave like a tensor, you need to compare it to two vectors to get a number, in the way I outlined above.
I would suggest that you focus on linear algebra vocabulary and notation. It is easy to get things confused without something solid to fall back on. I suggest reading through this cheap book, and working on the problems.
EDIT: Probably shouldn't link pirated copies of things.
This webpage has a solid list of recommended textbooks: https://mathblog.com/mathematics-books/
For Linear Algebra, Linear Algebra Done Right (3rd Ed.).
Linear Algebra can be of different levels of difficulty:
I keep referencing this site and keep referring back to it. I'm making my own plan, but I started with this as my template: http://www.beginnertriathlete.com/Scott%20Herrick/halfim/preparing_for_your_first_half_ir.htm
I bought these books this past weekend and I'm learning a lot from them:
http://www.amazon.com/Triathlete-Magazines-Complete-Triathlon-Book/dp/0446679283/ref=cm_lmf_img_7
http://www.amazon.com/The-Triathletes-Training-Bible-Friel/dp/1934030198/ref=cm_lmf_img_2
For Functional Analysis you need to be comfortable with Linear Algebra and Real Analysis.
There are tons of amazing books on either subject.
Matrix Analysis and Applied Linear Algebra Book and Solutions Manual by Carl D Meyer, for example, should leave you by the doorsteps of Functional Analysis.
A Linear Algebra Problem Book, by Paul Halmos is a great book to take your understanding of linear algebra to a deeper level. He has hints and complete solutions to all of his problems, and is known as a wonderful mathematical writer. Since linear algebra is vital in economics, this would be worthwhile. It also builds toward functional analytic ideas, which are used in very high level economics and differential equations.
Thanks! Link for anyone interested
If your priority is training for the Tri, a muscle building program like SL will not be very helpful.
You would be much better off following an endurance program that peaks on your event date. You still have a couple months to establish base and then another couple months added anaerobic and intervals.
Read this entire book- it will help you plan a good peak - http://www.amazon.com/Triathletes-Training-Bible-Joe-Friel/dp/1934030198/ref=dp_ob_title_bk
Yes. Halmos's Linear Algebra Problem Book takes this approach.
I'm not sure what kind of shape you're in, but I'm guessing that the ironman requires a lot more planning just to finish it. I'd suggest getting a copy of this book which will help you plan out and train for all three sports.
Depending on the area you're in, I'd suggest joining a club that does group worksouts (runs, rides, swims, etc). Very useful for all sorts of things, but especially for organized pool workouts. If you're in the DC area, I'll suggest (Team Z)[http://www.triteamz.com/], but I'm sure there are other teams out there.
The only one that comes to mind is Linear Algebra Problem Book by Halmos, but I'm not sure about how challenging it is.
There are some recommendations on Amazon :
>I find it ironic that my two favourite Linear Algebra texts are this book and the Axler, for they are exact opposites: Axler shuns determinants, and Shilov starts with them and builds much of his theory off them. However, there is no book I have found that has such a deep and clear exposition of determinants. The first chapter alone makes this book worth buying.
http://www.amazon.com/Linear-Algebra-Dover-Books-Mathematics/dp/048663518X/ref=sr_1_1?s=books&ie=UTF8&qid=1346872221&sr=1-1&keywords=linear+algebra
I would suggest this book for more advanced reading : http://www.amazon.com/gp/product/0415267994/ref=cm_cr_mts_prod_img
^ That book is really good. It starts with linear algebra topics and moves into functional analysis.
Can you run on the deck of the ship?
If you are already pretty fit (which I assume you are since you are in the Navy), you shouldn't have too much of an issue finishing an Oly. If you are shooting for a specific time goal you will be a bit more constrained however.
You have quite a bit of time until early summer so I would build up a strong aerobic base and maybe incorporate a bit of weights in for lower body and upper body. I would be careful with maximal weights at this point. Try to go for low weight and a lot of reps. Try to avoid putting on a ton of mass -- keep it lean.
Joe Friel writes some amazing books that you would find very interesting and helpful in structuring your plan. See the Triathlete's Training Bible.
I would say it is absolutely doable. Joe Friel says tris are a swim warm up, a bike race, and a jog to the finish. So you being a cyclist, yes. Yes you can do it.
Wow, do you go to some school where mathochism is cool? This is not a junior-level course in my academic worldview. It was not too too long ago that linear algebra was almost exclusively a graduate course. It was pushed down to the undergraduate level because of its extreme usefulness in ODEs and DSP, among other things. Undergraduates did not get that much smarter, instead the curriculum for linear algebra just got that much more streamlined. Your prof is either ignorant of or doesn't care about that evolution. If this is supposed to be a "regular" class, then you might voice a complaint to the chairperson of the department. Junior level courses usually are the introduction to mathematical rigor, not the launchpad for the study of Lie Algebras or other specialized areas. However if you are in an honors class or a hardcore mathematics school, you'll just have to strap in and enjoy the ride.
So here's some rope. All my references are old because I am old(ish). However, you can probably do better with keyword searches in Wikipedia and WolframAlpha based on your lecture notes. Do something like a mind map of the connections. The only thing you are missing online will be problems.
Go to your library and get Linear Algebra and Its Applications. I learned from an earlier edition of the book, but I can't imagine it getting worse. The people who hate the book are the ones who didn't do the exercises. If you stick with it, it is very cool and things start to build and just make sense. Strang is an excellent, excellent expositor, but you have to be a big picture person. He also tells you exactly what the core of the book is, The Fundamental Theorem of Linear Algebra. Grok that and linear algebra is your oyster, e.g., Gram-Schmidt will seem like an obvious thing. (And wouldn't you know, a reference on that Wikipedia page is to a paper by Strang on just that(pdf).)
If you can put up with older notation, you will find a lot in the famous book by Halmos, Finite Dimensional Vector Spaces.
A lot of this carries through to graduate algebra and functional analysis, so find whatever texts your graduate courses require and check their indices. From the above it sounds like your prof is trying to hit all the connections to other areas.
This next book will probably not help you, but it is just crazy enough to make me think you may find some of your professor's thoughts hidden there, Mathematical Physics by Geroch. You don't have time to learn category theory, but his exposition ends up at the spectral theorem, I seem to recall. Seeing another presentation of those powerful theorems might be illuminating. (It's a beautiful book, but I've never heard of it being used in a class.)
If you don't have MATLAB, get a (free?/cheap?) student edition and play with it for "real" examples of what you are doing. Going through the Theorem-Proof process never worked for me with things like linear algebra: Seeing how you can pull things apart and put them back together is what makes the power of linear algebra come alive and gives you some motivation.
The last piece of advice is not a guarantee, but has always worked for me when in a draconian course: Drill yourself on your old tests and quizzes and homework. When everyone is failing and the final comes around, chances are good (for various reasons, including pity and laziness) that the earlier exams are almost exactly recapitulated. Use your prof's office hours to go over the subtleties of the exam problems. If you are engaged with the material, the chances are good that he will extend the scope of discussion and pull in examples from the current lectures. That's a very handy insight to have.
If the notes of your class do make it online, please think of linking it back here. I'm curious as to how deep this course is since it is pretty wide.
Linear Algebra (Fourth Edition) by Stephen H. Friedberg
EDIT: I just realized that you already mentioned this book in your comments. I used this book in my upper level course too and it was a real treat.
Almost forgot to reply. Linear Algebra by Friedberg is one of the more mathematically rigorous texts I've seen for undergraduates. My school used it in the honors linear algebra course. I think you'll find that it covers most of what you need. Hope it helps (if you can find it at the library or something).
Personally, I would take the time to read them both. A strong linear algebra background will be very helpful in ML. Its especially useful if you want to expand out a little bit more into other areas of signal processing. Make sure you also spend some time getting a good background in probability and statistics.
EDIT: I haven't actually read Axler's book but me and some of my friends are partial to this book.
Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach, by Hubbard and Hubbard.
I would like to say as a physicist that this book contains everything I have ever needed to know save contour integration. Calculus, linear algebra, and differential forms are extremely well tied together, and extremely important concepts such as Lagrange multipliers are developed as applications of the core material. Amazing book.
A few Suggestions:
All this books can easily be found. Good luck.
I can't help you with professors, but back when I took linear algebra in 2010 I found Linear Algebra by Hoffman and Kunze to be very helpful.
Link to the US version.
For sale
STAT 215 - Engineering Statistics
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>Textbook: Probability and Statistics for Engineering and the Sciences
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>Authors: Devore
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>Edition: 6
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>ISBN-10: 0495557447
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>ISBN-13: 9780495557449
MATH 208 - Linear Algebra I
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>Textbook: Linear Algebra and Its Applications
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>Authors: David C. Lay
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>Edition: 3
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>ISBN-10: 0321287134
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>ISBN-13: 9780321287137
Want to buy
SP&M S 181 - Communication Theory
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>Textbook: Applying Communication Theory for Professional Life: A Practical Introduction
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>Authors: Dainton, Zelley
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>Edition: 2nd
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>ISBN-10: 141297691X
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>ISBN-13: 9781412976916
I haven't taken the class at my school, but I purchased this book on the recommendation of my math professor. I am really enjoying the book.
I think what I enjoy most about the author, David C. Lay, is that he provides lots of useful areas of application as well as many descriptive models. All the material is, in my opinion, fascinating.
Even if your degree program doesn't require it, which is a surprise to me, I would still take it or at least get a book and learn on your own. Linear Algebra is really eye opening, or at least it was to me. I am still learning, but I think that It would help you understand other individual branches of mathematics as well as mathematics as a whole.
EDIT: The book is not actually that expensive, you can find them way cheaper maybe I should just link to a cheaper version of the book...
For math you're going to need to know calculus, differential equations (partial and ordinary), and linear algebra.
For calculus, you're going to start with learning about differentiating and limits and whatnot. Then you're going to learn about integrating and series. Series is going to seem a little useless at first, but make sure you don't just skim it, because it becomes very important for physics. Once you learn integration, and integration techniques, you're going to want to go learn multi-variable calculus and vector calculus. Personally, this was the hardest thing for me to learn and I still have problems with it.
While you're learning calculus you can do some lower level physics. I personally liked Halliday, Resnik, and Walker, but I've also heard Giancoli is good. These will give you the basic, idealized world physics understandings, and not too much calculus is involved. You will go through mechanics, electromagnetism, thermodynamics, and "modern physics". You're going to go through these subjects again, but don't skip this part of the process, as you will need the grounding for later.
So, now you have the first two years of a physics degree done, it's time for the big boy stuff (that is the thing that separates the physicists from the engineers). You could get a differential equations and linear algebra books, and I highly suggest you do, but you could skip that and learn it from a physics reference book. Boaz will teach you the linear and the diffe q's you will need to know, along with almost every other post-calculus class math concept you will need for physics. I've also heard that Arfken, Weber, and Harris is a good reference book, but I have personally never used it, and I dont' know if it teaches linear and diffe q's. These are pretty much must-haves though, as they go through things like fourier series and calculus of variations (and a lot of other techniques), which are extremely important to know for what is about to come to you in the next paragraph.
Now that you have a solid mathematical basis, you can get deeper into what you learned in Halliday, Resnik, and Walker, or Giancoli, or whatever you used to get you basis down. You're going to do mechanics, E&M, Thermodynamis/Statistical Analysis, and quantum mechanics again! (yippee). These books will go way deeper into theses subjects, and need a lot more rigorous math. They take that you already know the lower-division stuff for granted, so they don't really teach those all that much. They're tough, very tough. Obvioulsy there are other texts you can go to, but these are the one I am most familiar with.
A few notes. These are just the core classes, anybody going through a physics program will also do labs, research, programming, astro, chemistry, biology, engineering, advanced math, and/or a variety of different things to supplement their degree. There a very few physicists that I know who took the exact same route/class.
These books all have practice problems. Do them. You don't learn physics by reading, you learn by doing. You don't have to do every problem, but you should do a fair amount. This means the theory questions and the math heavy questions. Your theory means nothing without the math to back it up.
Lastly, physics is very demanding. In my experience, most physics students have to pretty much dedicate almost all their time to the craft. This is with instructors, ta's, and tutors helping us along the way. When I say all their time, I mean up until at least midnight (often later) studying/doing work. I commend you on wanting to self-teach yourself, but if you want to learn physics, get into a classroom at your local junior college and start there (I think you'll need a half year of calculus though before you can start doing physics). Some of the concepts are hard (very hard) to understand properly, and the internet stops being very useful very quickly. Having an expert to guide you helps a lot.
Good luck on your journey!
Roman is great for what you want.
There is one more suggestion I can offer, which is at the level of learning habits and psychology: https://www.amazon.com/Mind-Numbers-Science-Flunked-Algebra/dp/039916524X
It's written for a much more popular audience than the earlier suggestions, but I still found it helpful.
The instinct for a lot of people is that when they get stuck, they think that the way forward is to isolate oneself to that problem and batter themselves at it until they solve it. The author does a good job explaining why this is almost always the wrong approach, and offers some psych-ish suggestions on better approaches. For example, she describes the difference between "diffuse" vs "focused" thinking, and how important it is to learn to switch between the two modes, so you don't get stuck performing focused thinking in the wrong area. Or how memory needs to be allowed to "chunk" so that it can form larger mental maps.
Good luck!
Edit to say, as for where homomorphisms are used, one cool application is that linear transformations (vector space homomorphisms) are a close analog of group homomorphisms. Having taken your group theory class, you may find something like this interesting? https://www.amazon.com/Linear-Algebra-Introduction-Mathematics-Undergraduate/dp/0387940995
Honestly, I think you should be more realistic: doing everything in that imgur link would be insane.
You should try to get a survey of the first 3 semesters of calculus, learn a bit of linear algebra perhaps from this book, and learn about reading and writing proofs with a book like this. If you still have time, Munkres' Topology, Dummit and Foote's Abstract Algebra, and/or Rudin's Principles of Mathematical Analysis would be good places to go.
Roughly speaking, you can theoretically do intro to proofs and linear algebra independently of calculus, and you only need intro to proofs to go into topology (though calculus and analysis would be desirable), and you only need linear algebra and intro to proofs to go into abstract algebra. For analysis, you need both calculus and intro to proofs.
You could consider starting with a book like Velleman's How to Prove It. It doesn't have to be that book, there are also free options online, but learning some logic and set theory from a book like that is a good way to figure out how to work with the other subjects you're working on.
Then, you could find a rigorous treatment of the subjects you want to learn. Something like Axler's Linear Algebra Done Right or Spivak's Calculus.
Learning math from textbooks like this is harder, but you end up with a better understanding of the math.
I've never taught the course, but a couple of my colleagues are very fond of Linear Algebra Done Wrong and would willingly teach from it if (1) the title wouldn't immediately turn students off of it and (2) the school would be okay with sacrificing some income from students having to purchase a book.
If you're curious, the book title is a play on the title of another well-known linear algebra book.
Linear algebra is about is about linear functions and is typically taken in the first or second year of college. College algebra normally refers to a remedial class that covers what most people do in high school. I highly recommend watching this series of videos for getting an intuitive idea of linear algebra no matter what book you go with. The book you should use depends on how comfortable you are with proofs and what your goal is. If you just want to know how to calculate and apply it, I've heard Strang's book with the accompanying MIT opencourseware course is good. This book also looks good if you're mostly interested in programming applications. A more abstract book like Linear Algebra Done Right or Linear Algebra Done Wrong would probably be more useful if you were familiar with mathematical proofs beforehand. How to Prove it is a good choice for learning this.
I haven't seen boolean algebra used to refer to an entire course, but if you want to learn logic and some proof techniques you could look at How to Prove it.
Most calculus books cover both differential and integral calculus. Differential calculus refers to taking derivatives. A derivative essentially tells you how rapidly a function changes at a certain point. Integral calculus covers finding areas under curves(aka definite integrals) and their relationship with derivatives. This series gives some excellent explanations for most of the ideas in calculus.
Analysis is more advanced, and is typically only done by math majors. You can think of it as calculus with complete proofs for everything and more abstraction. I would not recommend trying to learn this without having a strong understanding of calculus first. Spivak's Calculus is a good compromise between full on analysis and a standard calculus class. It's possible to use this as a first exposure to calculus, but it would be difficult.
It's aight. Just read linear algebra, and mv calculus. Maybe some statistical mechanics, read some thermo and kinetics. Atkins for kinetics and thermo, McQuarrie for stat mech. For linear algebra read get this. You'll still have to take classes on it, so it's cool. The worst you may have to do is take some UG classes to get up to speed.
I did an independent study of category theory from Goldblatt's Topoi: The Categorical Analysis of Logic. Having not studied category theory further than that, I can't offer much comparison, but I found it just barely accessible (which is generally about the best I hope for) and pretty cheap to boot.
Also, this book is a tough piece of work, for sure, but it's very helpful. It probably goes deeper than your class will, and may present ideas/methods in a different way, but if you grapple w/ this one, it'll really help you figure out L.A.
I used these lectures and skipped 23-28 and all of the review lectures. Though, you may want to review if there is any material in there that would be on the exam. I just ran out of time / got lazy towards the end. It helped me to buy the book and do homework assignments in the relevant chapters as I watched each video. It's not the same book used in the lectures, but for the most part it follows, and if it doesn't it was just out of order. The textbook is okay but is more or less the video lectures with the chalkboard diagrams and examples in print; there's not that much additional information. Doing practice problems is invaluable. Much of Math 415 is algorithmic.
That helps a little. I'm not too familiar with that world (I'm a physics major), but I took a look at a sample civil engineering course curriculum. If you like learning but the material in high school is boring, you could try self-teaching yourself basic physics, basic applied mathematics, or some chemistry, that way you could focus more on engineering in college. I don't know much about engineering literature, but this book is good for learning ODE methods (I own it) and this book is good for introductory classical mechanics (I bought and looked over it for a family member). The last one will definitely challenge you. Linear Algebra is also incredibly useful knowledge, in case you want to do virtually anything. Considering you like engineering, a book less focused on proofs and more focused on applications would be better for you. I looked around on Amazon, and I found this book that focuses on applications in computer science, and I found this book focusing on applications in general. I don't own any of those books, but they seem to be fine. You should do your own personal vetting though. Considering you are in high school, most of those books should be relatively affordable. I would personally go for the ODE or classical mechanics book first. They should both be very accessible to you. Reading through them and doing exercises that you find interesting would definitely give you an edge over other people in your class. I don't know if this applies to engineering, but using LaTeX is an essential skill for physicists and mathematicians. I don't feel confident in recommending any engineering texts, since I could easily send you down the wrong road due to my lack of knowledge. If you look at an engineering stack exchange, they could help you with that.
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You may also want to invest some time into learning a computer language. Doing some casual googling, I arrived at the conclusion that programming is useful in civil engineering today. There are a multitude of ways to go about learning programming. You can try to teach yourself, or you can try and find a class outside of school. I learned to program in such a class that my parents thankfully paid for. If you are fortunate enough to be in a similar situation, that might be a fun use of your time as well. To save you the trouble, any of these languages would be suitable: Python, C#, or VB.NET. Learning C# first will give you a more rigorous understanding of programming as compared to learning Python, but Python might be easier. I chose these three candidates based off of quick application potential rather than furthering knowledge in programming. This is its own separate topic, but my personal two cents are you will spend more time deliberating between programming languages rather than programming if you don't choose one quickly.
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What might be the best option is contacting a professor at the college you will be attending and asking for advice. You could email said professor with something along the lines of, "Hi Professor X! I'm a recently accepted student to Y college, and I'm really excited to study engineering. I want to do some rigorous learning about Z subject, but I don't know where to start. Could you help me?" Your message would be more formal than that, but I suspect you get the gist. Being known by your professors in college is especially good, and starting in high school is even better. These are the people who will write you recommendations for a job, write you recommendations for graduate school (if you plan on it), put you in contact with potential employers, help you in office hours, or end up as a friend. At my school at least, we are on a first name basis with professors, and I have had dinner with a few of mine. If your professors like you, that's excellent. Don't stress it though; it's not a game you have to psychopathically play. A lot of these relationships will develop naturally.
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That more or less covers educational things. If your laziness stems from material boredom, everything related to engineering I can advise on should be covered up there. Your laziness may also just originate from general apathy due to high school not having much impact on your life anymore. You've submitted college applications, and provided you don't fail your classes, your second semester will probably not have much bearing on your life. This general line of thought is what develops classic second semester senioritis. The common response is to blow off school, hang out with your friends, go to parties, and in general waste your time. I'm not saying don't go to parties, hang out with friends, etc., but what I am saying is you will feel regret eventually about doing only frivolous and passing things. This could be material to guilt trip yourself back into caring.
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For something more positive, try to think about some of your fun days at school before this semester. What made those days enjoyable? You could try to reproduce those underlying conditions. You could also go to school with the thought "today I'm going to accomplish X goal, and X goal will make me happy because of Y and Z." It always feels good to accomplish goals. If you think about it, second semester senioritis tends to make school boring because there are no more goals to accomplish. As an analogy, think about your favorite video game. If you have already completed the story, acquired the best items, played the interesting types of characters/party combinations, then why play the game? That's a deep question I won't fully unpack, but the simple answer is not playing the game because all of the goals have been completed. In a way, this is a lot like second semester of senior year. In the case of real life, you can think of second semester high school as the waiting period between the release of the first title and its sequel. Just because you are waiting doesn't mean you do nothing. You play another game, and in this case it's up to you to decide exactly what game you play.
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Alternatively, you could just skip the more elegant analysis from the last few paragraphs and tell yourself, "If I am not studying, then someone else is." This type of thinking is very risky, and most likely, it will make you unhappy, but it is a possibility. Fair warning, you will be miserable in college and misuse your 4 years if the only thing you do is study. I guarantee that you will have excellent grades, but I don't think the price you pay is worth it.
Cough
this is the book: http://www.amazon.com/Linear-Algebra-Edition-Stephen-Friedberg/dp/0130084514
There is no "one fastest" method to solving them.
Systems of equations are systematic, and it really depends on the problem. The only real way to learn about this is to take a course in Linear Algebra. That is all about systems of linear equations.
But these show up all of the time, here is what I usually do:
If I just need one of the 2-3 variables, Cramer's Rule is a good way to test solvability and extract a single value.
On normal 2x2 systems, I usually do a quick determinant/matrix inverse. Checks the rank as well as the det, and it is always going to work.
On 3x3 or higher systems, it depends. This is why Linear Algebra is important.
Supposedly Linear Algebra Done Right is a good book on the subject, so if you're interested there is one way. The book I used was A custom edition of this one. I thought it was very good as well.
Working through Griffiths is a good idea, but I strongly suggest working through an abstract linear algebra book before you do anything else. It will make your life much better. Doing some of Griffiths in advance might make your homework a bit easier, but you'll be repeating material when you could be learning new things. And learning real linear algebra will benefit you in pretty much every class.
I recommend this book as your primary text and this one for extra problems and and a second opinion.
I've been reviewing linear algebra recently and found that I like my old textbook much more now than when I took the course.
https://www.amazon.com/Linear-Algebra-4th-Stephen-Friedberg/dp/0130084514
Its not very good on visual intuition but there are a lot of examples. You could supplement it with the 3blue1brown series for that.
It covers a lot of the topics i needed to review for group theory. For example, it covers dual spaces and the transpose in the second chapter (it stresses invariant subspaces, projection operators, bilinear forms- essentials for group theory.). It's clear, concise and seems popular. One of the prof.s featured on Numberphile said he used it for his course. It might not be a good first linear algebra book for some people. But check it out.
I'm a huge fan of linear algebra. My favorite book for a theoretical understanding is this book. A pdf copy of the solutions manual can be found here.
Linear algebra by Friedberg, Insel and Spence
http://www.amazon.ca/Linear-Algebra-Edition-Stephen-Friedberg/dp/0130084514
Hoffman and Kunze
Strang's book is a fantastic resource for learning linear algebra but as you stated that your problem in you current text is the fact that it does not offer as much theory as you would like I am going to recommend another one of the MIT books Linear Algebra by Hoffman and Kunze I used this text book in my honors class and it is definitely not short on theory but you might want to keep you other textbook around for clarification on some issues as it can be quite opaque at some parts.
I’d recommend reading a text on linear algebra. Hoffman is pretty thorough: Linear Algebra (2nd Edition) https://www.amazon.com/dp/0135367972/ref=cm_sw_r_cp_api_oZkMAb4NH15B3
This text is the bible of linear algebra.
Grab a copy of this book from a local university or public library (or pay the massive price tag if you can afford it). It's a great text.
I think a rigorous course in linear algebra is the right place to start. Not only does the subject in some sense unify geometry and algebra, it's also necessary to understand it if you want to understand more advanced topics.
Strang is alright. Hoffman and Kunze is where it's really at.
What do you want to do, though? Is your goal to read math textbooks and later, maybe, math papers or is it for science/engineering? If it's the former, I'd simply ditch all that calc business and get started with "actual" math. There are about a million books designed to get you in the game. For one, try Book of Proof by Richard Hammack. It's free and designed to get your feet wet. Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand/Polimeni/Zhang is my favorite when it comes to books of this kind. You'll also pick up a lot of math from Discrete Math by Susanna Epp. These books assume no math background and will give you the coveted "math maturity".
There is also absolutely no shortage of subject books that will nurse you into maturity. For example, check out [The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs by Grinberg](https://www.amazon.com/Real-Analysis-Lifesaver-Understand-Princeton/dp/0691172935/ref=sr_1_1?ie=UTF8&qid=1486754571&sr=8-1&keywords=real+analysis+lifesaver() and Book of Abstract Algebra by Pinter. There's also Linear Algebra by Singh. It's roughly at the level of more famous LADR by Axler, but doesn't require you have done time with lower level LA book first. The reason I recommend this book is because every theorem/lemma/proposition is illustrated with a concrete example. Sort of uncommon in a proof based math book. Its only drawback is its solution manual. Some of its proofs are sloppy, messy. But there's mathstackexchange for that. In short, every subject of math has dozens and dozens of intro books designed to be as gentle as possible. Heck, these days even grad level subjects are ungrad-ized: The Lebesgue Integral for Undergraduates by Johnson. I am sure there are such books even on subjects like differential geometry and algebraic geometry. Basically, you have choice. Good Luck!
I used Lay's book in school as well, found it pretty easy to understand, though the professor was a baller shot-caller. I agree though, sometimes it isn't straightforward enough. Good practice problems though: http://www.amazon.com/Linear-Algebra-Applications-Updated-CD-ROM/dp/0321287134/ref=sr_1_2?s=books&ie=UTF8&qid=1373219520&sr=1-2&keywords=linear+algebra+and+it%27s+applications
For algorithms, I would recommend checking out Skiena's "Algorithm Design Manual." One of the cool features are his "War Stories" which give various examples of how the author used and adapted algorithms to solve real-world problems.
For linear algebra, I haven't read it myself but you might try Lay's "Linear Algebra and Its Applications" which probably will have more of a focus on applications than the titles mentioned by KolmogorovTuring.
Linear Algebra And Its Applications by Lay. I think I'll download the Kindle apps on my laptop and iPhone and see if the sample from the book renders decently there.
The increased portability would be a godsend, but I'm not sure if the marginal cost of going digital is justified yet.
I don't have suggestions on a lecture series, but this is a pretty common book.
I keep seeing this book recommended in a lot of places. How is it different from the one by Axler and one by Roman?
I'm recently going through "Linear Algebra" by Robert Valenza.
It is very compact and elegant. I find it excellent for an abstract introduction to Linear Algebra.
I'd love to hear your opinion on it, if you're familiar with it.
Don't skip proofs and wrestle through them. That's the only way; to struggle. Learning mathematics is generally a bit of a fight.
It's also true that computation theory is essentially all proofs. (Specifically, constructive proofs by contradiction).
You could try a book like this: https://www.amazon.com/Book-Proof-Richard-Hammack/dp/0989472108/ref=sr_1_1?ie=UTF8&qid=1537570440&sr=8-1&keywords=book+of+proof
But I think these books won't really make you proficient, just more familiar with the basics. To become proficient, you should write proofs in a proper rigorous setting for proper material.
Sheldon Axler's "Linear Algebra Done Right" is really what taught me to properly do a proof. Also, I'm sure you don't really understand Linear Algebra, as will become very apparent if you read his book. I believe it's also targeted towards students who have seen linear algebra in an applied setting, but never rigorous and are new to proof-writing. That is, it's meant just for people like you.
The book will surely benefit you in time. Both in better understanding linear algebra and computer science classics like isomorphisms and in becoming proficient at reading/understanding a mathematical texts and writing proofs to show it.
I strongly recommend the second addition over the third addition. You can also find a solutions PDF for it online. Try Library Genesis. You don't need to read the entire book, just the first half and you should be well-prepared.
As others mentioned, it is very hard to make progress learning programming without using a computer (think of reading about driving without ever driving a car). Instead, get yourself excited about science and computer science:
Science:
Computer Science (actually math, but this will help change the way you think to be more analytical, and will be useful for programming, vector graphics, etc.):
I think linear algebra is a much more interesting topic without getting bogged down in matrix computations, such as what Axler does with Linear Algebra Done Right. That's just my opinion I suppose.
Hello! I'm interested in trying to cultivate a better understanding/interest/mastery of mathematics for myself. For some context:
 
To be frank, Math has always been my least favorite subject. I do love learning, and my primary interests are Animation, Literature, History, Philosophy, Politics, Ecology & Biology. (I'm a Digital Media Major with an Evolutionary Biology minor) Throughout highschool I started off in the "honors" section with Algebra I, Geometry, and Algebra II. (Although, it was a small school, most of the really "excelling" students either doubled up with Geometry early on or qualified to skip Algebra I, meaning that most of the students I was around - as per Honors English, Bio, etc - were taking Math courses a grade ahead of me, taking Algebra II while I took Geometry, Pre-Calc while I took Algebra II, and AP/BC Calc/Calc I while I took Pre-Calc)
By my senior year though, I took a level down, and took Pre-Calculus in the "advanced" level. Not the lowest, that would be "College Prep," (man, Honors, Advanced, and College Prep - those are some really condescending names lol - of course in Junior & Senior year the APs open up, so all the kids who were in Honors went on to APs, and Honors became a bit lower in standard from that point on) but since I had never been doing great in Math I decided to take it a bit easier as I focused on other things.
So my point is, throughout High School I never really grappled with Math outside of necessity for completing courses, I never did all that well (I mean, grade-wise I was fine, Cs, Bs and occasional As) and pretty much forgot much of it after I needed to.
Currently I'm a sophmore in University. For my first year I kinda skirted around taking Math, since I had never done that well & hadn't enjoyed it much, so I wound up taking Statistics second semester of freshman year. I did okay, I got a C+ which is one of my worse grades, but considering my skills in the subject was acceptable. My professor was well-meaning and helpful outside of classes, but she had a very thick accent & I was very distracted for much of that semester.
Now this semester I'm taking Applied Finite Mathematics, and am doing alright. Much of the content so far has been a retread, but that's fine for me since I forgot most of the stuff & the presentation is far better this time, it's sinking in quite a bit easier. So far we've been going over the basics of Set Theory, Probability, Permutations, and some other stuff - kinda slowly tbh.
 
Well that was quite a bit of a preamble, tl;dr I was never all that good at or interested in math. However, I want to foster a healthier engagement with mathematics and so far have found entrance points of interest in discussions on the history and philosophy of mathematics. I think I could come to a better understanding and maybe even appreciation for math if I studied it on my own in some fashion.
So I've been looking into it, and I see that Dover publishes quite a range of affordable, slightly old math textbooks. Now, considering my background, (I am probably quite rusty but somewhat secure in Elementary Algebra, and to be honest I would not trust anything I could vaguely remember from 2 years ago in "Advanced" Pre-Calculus) what would be a good book to try and read/practice with/work through to make math 1) more approachable to me, 2) get a better and more rewarding understanding by attacking the stuff on my own, and/or 3) broaden my knowledge and ability in various math subjects?
Here are some interesting ones I've found via cursory search, I've so far just been looking at Dover's selections but feel free to recommend other stuff, just keep in mind I'd have to keep a rather small budget, especially since this is really on the side (considering my course of study, I really won't have to take any more math courses):
Prelude to Mathematics
A Book of Set Theory - More relevant to my current course & have heard good things about it
Linear Algebra
Number Theory
A Book of Abstract Algebra
Basic Algebra I
Calculus: An Intuitive and Physical Approach
Probability Theory: A Concise Course
A Course on Group Theory
Elementary Functional Analysis
I bought a copy of Dover's Linear Algebra (Border's Blowout) which I plan to go through after I finish A Book of Abstract Algebra.
I feel like I have a long way to go to get anywhere. :S
I meant it quite literally, something along the lines Linear Algebra by Georgi E. Shilov, but less rigorous.
> Leslie Ballentine's book
these
https://www.amazon.co.uk/Quantum-Mechanics-Development-Leslie-Ballentine/dp/9810241054
https://www.amazon.co.uk/Linear-Algebra-Dover-Books-Mathematics/dp/048663518X/ref=sr_1_1?s=books&ie=UTF8&qid=1542818674&sr=1-1&keywords=linear+algebra+Halmos
You're right, I just picked up this Linear Algebra book by Lang and finishing up the first chapter on vectors. Matrices are up next. I am glad this question popped up because now I have a good mnemonic for "consistent" and "linearly dependent" both of which seem to be a big deal in Linear Algebra :)
Thank you.
https://vimeo.com/ may have some, try to search "linear algebra " there. And sometimes books are good too. Search on amazon.
https://www.amazon.com/Matrix-analysis-applied-linear-algebra/dp/0898714540/ref=sr_1_15?ie=UTF8&qid=1511664543&sr=8-15&keywords=linear+algebra+books
Like 50 on amazon but could also try Abebooks and see if there's a cheaper used or international copy.
I don't know much about AI, though I do know that (there's a theme, here) linear algebra gets a starring role. So, if you're currently enjoying linear algebra, continue with that. Axler is frequently recommended, if you want a textbook to go through.
After that it's really up to you what you want to go for next, since you have many paths available. Sipser is a great intro to theoretical CS, but, again, don't spend $200 on it. Try to find it in a library, or use something like this to find a much cheaper international edition.
Edit: Forgot to mention, CLRS is the standard for algorithms, but I'm not sure how useful it is as a primary source for learning. Maybe try to borrow a copy to see if you like it.
For Linear Algebra there's very in depth Matrix Analysis and Applied Linear Algebra Book and Solutions Manual. Great for self-teaching since it's very detailed, user-friendly and comes with solutions manual. If you are somewhat comfortable with some abstractness, there's a text called Linear Algebra Done Wrong by Sergei Treil. It's excellent and free.
Late, but here are undergrad books on the subject: geometric algebra, geometric calculus.
A grad-type book that has both and their applications to physics would be this one
I'm currently learning the geometric algebra undergrad book. It's a good read so far, and the author keeps up with book errors.
Unfortunately not really. I would like to make an interactive website version at some indefinite future time.
There is http://www.xtec.cat/~rgonzal1/treatise2.htm but it ramps up in difficulty / expected background faster than I had hoped.
There’s Hestenes’s book New Foundations for Classical Mechanics, but again somewhat tough, and very physics focused.
There’s Alan MacDonald’s book https://amzn.com/1453854932 but it’s too coordinate-focused for my taste.
http://www.amazon.com/The-Triathletes-Training-Bible-Friel/dp/1934030198/ref=sr_1_sc_1?ie=UTF8&qid=1398039218&sr=8-1-spell&keywords=triathalets+training+bible
Totally worth it.
Sorry, I went on vacation and totally blanked about posting these for you!
Anyway, here are some books
Linear Algebra Done Right (Undergraduate Texts in Mathematics) https://www.amazon.com/dp/3319110799/ref=cm_sw_r_cp_api_1L8Byb5M5W9D3
This one is actually for analysis but depending on your appetite, it might help greatly with the proof side of your class. You can buy it here: Counterexamples in Analysis (Dover Books on Mathematics) https://www.amazon.com/dp/0486428753/ref=cm_sw_r_cp_api_GS8BybQWYBFXX
But there's also a PDF hosted here: http://www.kryakin.org/am2/_Olmsted.pdf
A cool Mark Allen article, to get you in the mood
THE ONE TRUE BOOK (Accept no imitations)
Hal Higdon's Site
Joe Friel's book about how too piss on your bike seat, I mean, complete a triathlon
There is a lot in the way of resources for new triathletes these days. For your first tri, grab a free training plan online that matches where you are now. Read Beginner Triathlete in your free time; it's a fantastic resource, and I still refer back to its articles all the time. Train your butt off. You don't need to buy a sweet road bike up front, though you sound like you're pretty sure that you want to get into this stuff.
Feel free to skimp on some of the gear for your first race. No one wants to find out that they dislike triathlon after dumping $3k on tri gear. You can race on an old bike with platform pedals. Unless it's really cold, you don't need a wetsuit. The first race is where you truly find out if this is the sport for you. EDIT: Someone mentioned a bike fit. If you're riding an old bike, Competitive Cyclist's Bike Fit Calculator will get you pretty darn close--good enough to get through your first race. Use the road calculator mode if you don't have aerobars off the bat.
After you finish your first race, sit down and think about what you liked, what you did well with, what needs improvement. Get Joe Friel's Triathlete's Training Bible, read it cover to cover. Read it again. Figure out your long-term training plan for the rest of that season. If you start your base training in the winter/early spring and pick an early first race, you can get a full season of sprints and/or Olympics in.
Look for a triathlon club in your area or find a coach or drag a friend into the insanity of triathlon; the camaraderie is priceless in keeping your spirits up during long seasons packed full of hard training and races.
As far as spending money on triathlon "stuff" goes: Remember during your first couple seasons that gadgets and gizmos and aero gear are great, but what really makes the difference is eating well and training hard.
After that, the gear that makes your races more comfortable is the best place to spend your money (tri shorts if you don't them, cycling kit and proper running shorts for training). Then, points of contact with the bike and pool "toys" will improve your efficiency and form (new bike w/ fit if req'd, clipless pedals, shoes, aerobars, pull buoy, kickboard, fins, paddles... a bike computer probably fits in here, as well). Beyond that, you're at a wetsuit and then the "extras" like aero helmet, race wheels, power meters, GPS, HRM, tri bike, speedsuits, etc., etc. That's the approximate map for spending in my book, anyhow. There's practically no limit to the amount of stuff you can buy for triathlon, and as you train more, you'll know what needs to come next.
Introduction to Linear Algebra by Serge Lang.
http://www.amazon.com/Introduction-Linear-Algebra-Serge-Lang/dp/3540780602
Or Introduction to Linear Algebra by Gilbert Strang
http://www.amazon.com/Introduction-Linear-Algebra-Fourth-Edition/dp/0980232716
I have not used the Strang book, but I here it is all right for non-mathematicians.
Buy a training plan off amazon and follow it rigorously.
Something like
this
or this
Strang's Linear Algebra.
Here's a better Linear Algebra textbook (assuming that course is the context): http://www.amazon.com/Linear-Algebra-Applications-Updated-CD-ROM/dp/0321287134