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So, my math was teacher talking recently with me and a few other of the math students at my high school who do math competitions. We were complaining that our school doesn't teach any math beyond Calc BC, while many other schools have at least MVC. He said if our school were to ever have a more advanced class, it would be linear algebra. I looked in to linear algebra, since he said it was pretty easy, and found this book online: the book's page. Be warned, the pdf is quite large (465pages).
I've covered the first chapter (wimpy, I know) but I already have a few questions. Since I don't want to tell my math teacher that I'm studying this yet while I'm only done with the first (pretty basic) chapter, I was hoping that maybe I could put some questions on this blog when I have them.
First question: Is Gauss' method really that important? It seems like these systems of equations could be often solved much faster using simple matrix division, not to mention the ability to simply plug matrices into my calculator. I know that systems with non-specific solutions will still require some manipulation by hand, and that these solutions are probably pretty important since from what I can tell they could yield planes and other shapes in 3rd to xth dimensional spaces, but would these even require Gauss' method? It seems like substitution would be very useful in simplifying some systems, and from what I can tell Gauss' method really only uses elimination.
Second Question: Will I need to know any calculus beyond simple derivatives and integrals? The book description says no, but if I were to take a typical, more challenging linear algebra course at a well regarded university, would I need to know something more? It seems that it could at least be important to take the integral of some shapes to find volumes.
Third and final question: Will most universities let me test out of this course in some way? Since I'm interested in mathematics and will likely be going into a STEM field, I want to go in to uni with Calc, MVC, DiffE, and Linear Algebra out of the way to take more theoretical and difficult courses (I guess I would start with Discrete Math?). As much as it could boost my GPA, I don't want to take these math classes over again.
Thanks for your time and if you think the textbook is trash or have a recommendation I'm all ears.
Oh yeah, forgot to mention that I'm also motivated by a desire to do well in the Putnam competition in uni. I'm preparing for it by taking the above courses as well as, at first, working through AMC, AIME, and USAMO problems. If I'm missing a crucial course (I was thinking possibly set theory?) I'd really like to know.
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You should know how to calculate 154 times 74 on paper even if you always do it using a calculator. Same goes for Gauss's method. There's no point in drilling it for speed or anything like that but you should understand how it works. It's not that hard either, it's exactly the same as solving a system of linear equations by eliminating variables. Except you don't write down the name of variable and insert 0s as needed.
I'm guessing by matrix division you mean multiplying by the inverse. For most larger matrices you do calculate inverses by Gaussian elimination
You don't need any calculus to learn linear algebra. Maybe the book includes a lot of examples from calculus or something. OTOH, multivariable calculus and differential equations do use ideas from linear algebra.
Can't comment on the other questions though.
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By matrix division I do mean multiplying by the inverse, but I don't mean writing down a matrix and solving it by hand. I'd rather just use Gauss's method. I have a graphing calculator with matrix capabilities so I can just define a matrix A that is x by x , a matrix B that is x by 1 , and multiply the inverse of A by B to get a specific solution very quickly. To me this seems much faster than doing a whole bunch of elimination, putting things in echelon form, and then continually substituting values for variables.
I'm not bad at arithmetic. Hell, I am heavily involved in academic competitions in which I have to do a lot of arithmetic pretty quickly. I don't, however, want to do a lot of arithmetic that will only slow me down.
Thank you for your help! I'm glad I won't have to learn any more calculus.
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When I learned linear algebra in high school I learned from two sources:
First, I looked through Gilbert Strang's MIT OCW videos (look them up) and solved some of MIT's homework problem. Then I read Serge Lang's linear algebra book and did all the proofs from it. I highly recommend Lang's book if you are familiar with proofs from USAMO work. Some of my friends also like the book "Linear Algebra Done Right" by Axler.
There are two ways to learn linear algebra: the engineering way which is full of matrices and the mathematics way which introduces the concept of a vector space and hardly touches explicit matrices.
Regarding skipping linear algebra, any of the better schools will let you skip any math course you want to skip. They may not recommend it, but if you want to walk into algebraic geometry on the first day they will let you. The phrase for this is "enough noose to hang yourself."
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But how would you get the inverse? If your graphing calculator is doing it for you then that explains why its so easy; but otherwise you would need to get matrix A into identity matrix form while performing the same operations to transform an associated identity matrix into the inverse, which is basically the same as echelon row reduction. All of this is covered much, much later in your book, but it looks like you already know!
So either way you would have to go through that echelon row reduction, but the only difference is that with your method there is an extra step of multiplying (A inverse) with B
Also only some matrices have inverses - they have to be square (n x n) and have linearly independent rows/columns, so your method won't always yield a specific solution if you're working with an m x n matrix (where m does not equal n).
Just to be clear there shouldn't be any substitution required except for the trivial observation of what the results are, which you can read off of the reduced matrix. You should be only reducing one matrix, and the answers will be apparent at the end of that reduction without any substitution required.
As far as I'm aware, from my linear algebra course you can only get parametric (infinite) solutions using row reduction, although there may be more advanced ways taught in higher level courses. But I'm pretty sure they move on from linear algebra to non-linear cases. Gaussian elimination is simply a fast way to determine whether equations are linearly independent or dependent, while also providing you with any possible solutions to those equations.
And in my course (which was the more advanced version) we only required basic calculus and differentiation when we were dealing with maps from one vector space to another, which you will learn later. Basically you just integrate/differentiate simple polynomials.
I hope I didn't forget anything crucial or made any errors! I'm in Canada so I don't know the answer to your last question. It was an important prerequisite in my university (Toronto) for more advanced math courses, but I think you could get a transfer credit allowing you to skip that course. There is quite a lot you need to learn in advanced linear algebra beyond elementary matrix multiplication or row reduction. It gets complicated when they start talking about maps, isomorphisms...or maybe I fell behind  . Either way if you're considering putnam, I'm sure you'll find it quite nice and easy.
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On May 16 2013 07:15 Chocolate wrote:So, my math was teacher talking recently with me and a few other of the math students at my high school who do math competitions. We were complaining that our school doesn't teach any math beyond Calc BC, while many other schools have at least MVC. He said if our school were to ever have a more advanced class, it would be linear algebra. I looked in to linear algebra, since he said it was pretty easy, and found this book online: the book's page. Be warned, the pdf is quite large (465pages).I've covered the first chapter (wimpy, I know) but I already have a few questions. Since I don't want to tell my math teacher that I'm studying this yet while I'm only done with the first (pretty basic) chapter, I was hoping that maybe I could put some questions on this blog when I have them. First question: Is Gauss' method really that important? It seems like these systems of equations could be often solved much faster using simple matrix division, not to mention the ability to simply plug matrices into my calculator. I know that systems with non-specific solutions will still require some manipulation by hand, and that these solutions are probably pretty important since from what I can tell they could yield planes and other shapes in 3rd to xth dimensional spaces, but would these even require Gauss' method? It seems like substitution would be very useful in simplifying some systems, and from what I can tell Gauss' method really only uses elimination. Second Question: Will I need to know any calculus beyond simple derivatives and integrals? The book description says no, but if I were to take a typical, more challenging linear algebra course at a well regarded university, would I need to know something more? It seems that it could at least be important to take the integral of some shapes to find volumes. Third and final question: Will most universities let me test out of this course in some way? Since I'm interested in mathematics and will likely be going into a STEM field, I want to go in to uni with Calc, MVC, DiffE, and Linear Algebra out of the way to take more theoretical and difficult courses (I guess I would start with Discrete Math?). As much as it could boost my GPA, I don't want to take these math classes over again. Thanks for your time and if you think the textbook is trash or have a recommendation I'm all ears. Oh yeah, forgot to mention that I'm also motivated by a desire to do well in the Putnam competition in uni. I'm preparing for it by taking the above courses as well as, at first, working through AMC, AIME, and USAMO problems. If I'm missing a crucial course (I was thinking possibly set theory?) I'd really like to know.
Gaussian elimination is important in the sense that addition is important. It sets up the basics for later manipulations. You think you can just plug and chug everything (esp. in numerical computing)? No way. Inverse/substitution is naively decent, but carefully planned Gaussian elimination is much more efficient and less-error prone. Also, you can't do determinant manipulation if you don't know how/why Gaussian elim works.
You won't need much. The derivatives and integrals will be useful for infinite series stuff with matrices/exponential decomposition later.
Most universities shouldn't let you test out, since linalg is just so important in everything mathematics related at the higher level. Talk with your dept head if you really want to do something about it, but you better have the chops (as in, more than just being able to do formula shits like singular value decomposition or Gram-Schmidt...proofs are key).
Strang is decent, has a few errors. Lay is the standard HS book. Axler is nice, more abstract/less computational than a lot. It's one of the few books that doesn't barrel towards Gram-Schmidt as a conclusion.
If you're concerned about Putnam as a high priority, forget about learning linalg straight off the book. Just keep solving random linalg problems while struggling like a mofo looking through AoPS links...basic linear algebra (ESPECIALLY properties of trace/determinant/symmetric/triangular matrices) should be able to get you through 99% of Putnam problems. Although I'm kinda worried about your prospect at Putnam if you're just now getting into AMC/AIME/USAMO....
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Gauss's Method is Gaussian Elimination right? GE gives you an algorithm to determine the rank of the matrix (which isn't obvious by inspection in most matrices). In regular algebra you'd ask questions like, 2x=3 solve for x and you'd say x=3/2. Now in linear algebra I can ask you Ax=b, does x exist? If x exists, is it the unique solution? If it's not the unique solution, what does the space of solutions look like? What are the possible values of x? The rank gives you an answer to (some) these questions and you find the rank using GE. I mean, it also gives you an easy way to solve Ax=b by inspection but that's not the point.
This is the point, GE illustrates in some the sense the main idea that makes linear algebra work. We can manipulate a matrix into certain forms without altering the fundamental properties of the matrix and in these nice forms we can extract a bunch of information that makes solving the system easy (this is maximally vague lol).
2) Nope, no calculus necessary (although there is interplay of the two and some really nice applications but this is true for linear algebra and just about every branch of mathematics)
3) Usually you take diff eq and multi at a CC and they count that credit. If you just self study, they aren't going to let you skip all these classes based on just your word, you'll have to prove yourself in a course or two first. This isn't a bad thing; in fact, you'll probably gain a lot by taking them even if you're already familiar with the concepts. Self study can make you proficient in vocabulary and problem solving but, especially since you're just starting out with math, its very easy to miss the big picture. The why this stuff matters more than that stuff even though they're both in the chapter. How this material is a generalization of past material and etc. A teacher doesn't just know the material, they know how it all fits together
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Write a linear algebra calculator in C++ or Java or something, you'll quickly see why you'd rather do Gaussian elimination than take the inverse.
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On May 16 2013 16:01 jalstar wrote:
Write a linear algebra calculator in C++ or Java or something, you'll quickly see why you'd rather do Gaussian elimination than take the inverse.
And learning how to code is going to be more useful to you in the real world than knowing Linear Algebra in the long run. Math is obviously really important from a conceptual point of view, but from a practical point of view, programming is your lifeblood in most STEM fields.
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United States10328 Posts
On May 16 2013 16:01 jalstar wrote:
Write a linear algebra calculator in C++ or Java or something, you'll quickly see why you'd rather do Gaussian elimination than take the inverse.
I thought the computationally easiest way to take the inverse is to do Gaussian elimination anyway? haha
And @corpuscle: I think being comfortable with matrices/their manipulations is pretty important in a number of algorithms (graphics; optimization; fast fourier transform, etc). But I suppose you can learn how to code without knowing anything about algorithms 
Back to OP's questions:
1. Yes, as explained by lots of people.
2. Calculus is mostly orthogonal to linear algebra (ha, ha).
3. Probably, since it's an "intro" class.
On the textbook: It doesn't actually cover that much, but it might be a good intro? Didn't look at it closely enough to be sure, but it seems like it's trying to introduce some "mathematically rigorous things" while sticking to the "matrix" approach. Unless you think you're going to be doing theory, I'd recommend a more computational book tbh; you usually don't have to think much about vector spaces unless you're doing math or theoretical physics, and can get away with just matrix manipulation and some basic notions (I guess invertibility / diagonalization / eigenvectors are probably still important). If you'd like to do theory, this book might be ok, though I've heard lots of recommendations for Axler's Linear Algebra Done Right (which tries its darndest to avoid matrices for as long as possible!)
Personally, I learned linear algebra in a sort of "hybrid" way (really, as a subset of my abstract algebra course), and as a result I only sort of know the theory and can't compute very well  Probably more my fault than the course's, but I'd still recommend taking some time to pursue either the computational or the theoretical side in some detail.
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On May 17 2013 00:35 corpuscle wrote:Show nested quote +On May 16 2013 16:01 jalstar wrote:
Write a linear algebra calculator in C++ or Java or something, you'll quickly see why you'd rather do Gaussian elimination than take the inverse. And learning how to code is going to be more useful to you in the real world than knowing Linear Algebra in the long run. Math is obviously really important from a conceptual point of view, but from a practical point of view, programming is your lifeblood in most STEM fields.
Actually, if you know C/C++/Python/Matlab you're pretty much done, but more math is always useful in research
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EPT
