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I'm majoring in mathematics and currently enrolled in Linear Algebra. It's very different, but I like it (I think). My question is this: What doors does this course open? (I saw a post about Linear Algebra being the foundation for Applied Mathematics -- but I like doing math for the sake of math, not so much the applications.) Is this a stand-alone class, or will the new things I'm learning come into play later on?

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Well sure. Functional analysis, differential geometry, representation theory... even graph theory makes extensive use of linear algebra, and especially of matrices and eigenvalues. In fact, apart from set theory, I can't really think of a branch of math that doesn't build on linear algebra! – goblin Mar 14 at 6:19
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Literally Everything... – frogeyedpeas Mar 14 at 6:23
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@goblin: I suggest posting your answer as a proper one instead of a comment. – Martin Argerami Mar 14 at 6:26
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The basic idea of the infinitesimal calculus is that, if you look at anything close enough, it becomes linear. Of course this is only mostly true, but it means that theoretically and practically, linear algebra shows up in everything requiring derivatives. Free abelian groups are just an application of linear algebra. Graph theorey relies on linear algebra. Yes, it shows up everywhere. Even fractals, which don't get linear as you look closer, need linear algebra to describe themselves. – Steven Gregory Mar 14 at 6:27
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It's also also most students' first exposure to what mathematicians call algebra, and you'll see instances of quite a few good and broadly useful ideas: bases as generating sets, change-of-basis as conjugation, kernels, subspaces and quotients, and the list goes on! – pjs36 Mar 14 at 6:40
up vote 68 down vote accepted

Linear Algebra is indeed one of the foundations of modern mathematics. There are a lot of things which use the language and tools developed in linear algebra:

  • Multidimensional Calculus, i.e. Analysis for functions of many variables, i.e. vectors (for example, the first derivative becomes a matrix)

  • Differential Geometry, which investigates structures which look locally like a vector space and functions on this.

  • Functional Analysis, which is essentially linear algebra on infinite-dimensional vector spaces which is the foundation of quantum mechanics.

  • Multivariate Statistics, which investigates vectors whose entries are random. For instance, to describe the relation between two components of such random vector, one can calculate the correlation matrix. Furthermore, one can apply a technique called singular value decomposition (which is close to calculating the eigenvalues of a matrix) to find which components are having a main influence on the data.

  • Tagging on to the Multivariate Statistics and multidimensional calculus, there are a number of Machine Learning techniques which require you to find a (local) minimum of a nonlinear function (the likelihood function), for example for neural nets. Generally speaking, on can try to find the parameters which maximize the likelihood, e.g. by applying the gradient descent method, which uses vectors arithmetic. (Thanks, frogeyedpeas!)

  • Control Theory and Dynamical Systems theory is mainly concerned with differential equations where matrices are factors in front of the functions. It helps tremendously to know the eigenvalues of the matrices involved to predict how the system will behave and also how to change the matrices in front to make sure the system behaves like you want it to - in Control Theory, this is related to the poles and zeros of the transfer function, but in essence it's all about placing eigenvalues at the right place. This is not only relevant for mechanical systems, but also for electric engineering. (Thanks, Look behind you!)

  • Optimization in general and Linear Programming in particular is closely related to multidimensional calculus, namely about finding minima (or maxima) of functions, but you can use the structure of vector spaces to simplify your problems. (Thanks, Look behind you!)

On top of that, there are a lot of applications in engineering and physics which use tools of linear algebra and the fields listed above to solve real-world problems (often calculating eigenvalues and solving differential equations).

In essence, a lot of things in the mathematical toolbox in one variable can be lifted up to the multivariable case with the help of Linear Algebra.

Edit: This list is by no means complete, these were just the topics which came to my mind at first thought. Not mentioning a particular field doesn't mean that this field irrelevant, but just that I don't feel qualified to write a paragraph about it.

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Adding to the stats point, Machine Learning has been nothing but Lin Alg for me :) – frogeyedpeas Mar 14 at 6:30
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Control theory, signal processing, optimization, stochastic processes... – Look behind you Mar 14 at 6:44
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dynamical systems theory – Nathaniel Mar 14 at 11:39
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Numerical analysis often boils down to sparse matrix calculations. – lisyarus Mar 14 at 13:42
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Cryptography!!! – vojta Mar 15 at 8:07

By now, we can roughly say that all we fully understand in Mathematics is linear.

So I guess Linear Algebra is a good topic to master.

Both in Mathematics and in Physics you tend bring the problem to a Linear problem and then solve it with Linear Algebras techniques. This happens in Algebra with linear representations, in Functional Analysis with the study of Hilbert Spaces, in Differential Geometry with the tangent spaces, and so on almost in every field of Mathematics. Indeed I think there should be at least 3 courses of Linear Algebra (undergraduate, graduate, advanced), everytime with different insights on the subject.

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Put another way: we approximate a nonlinear reality with a more easily imagined linear simulacrum. – J. M. Mar 14 at 18:18
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In mathematics, we divide problems into linear problems and non-linear problems. This is like dividing the universe into bananas and not-bananas. – Yakk Mar 14 at 18:27
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+1 for the line "all we fully understand in mathematics is linear." – Placidia Mar 14 at 19:09
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well, If you fully understand Number Theory be my guest... I would have few million-dollars-questions to ask you... :D :D :D – Dac0 Mar 14 at 21:12
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@leftaroundabout Indeed linear algebra is different. If you study mathematics for a while you know that if you have a problem (in almost every context) is a common first practice to find some kind of way to reconduct you problem to a linear problem or your structure to a linear structure. Actually there's a reason for that, I'll let you guess what is... – Dac0 Mar 16 at 12:57

Excellent answers above. I would just like to add:

  • Computer graphics. As soon as you scale, rotate and/or translate an object on a computer screen, that is Linear Algebra in action. Computer graphics is an academic field that depend heavily on Linear Algebra.
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This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review – choco_addicted Mar 15 at 7:55
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Considering I have had classes in Computer Graphics (where we used Linear Algebra), that CG is an academic field, and that I have used LA professionally in that particular field, I strongly disagree with you. People above already mentioned many important fields that build on and/or use LA, and I saw no reason to repeat their excellent posts, but no-one mentioned CG. It would be folly of me to comment on each post above just to say "Do not forget CG". I made the judgement call it was best to mention CG in this way instead. – Michael Karnerfors Mar 15 at 8:00
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+1 Whenever a curious teenage student asks me "what is linear algebra, and why would I care?", I often root it in the real world by citing its use in graphics. – zahbaz Mar 15 at 17:40

Mathematics are all about building theorems and properties from a set of ground hypothesis.

Linear algebra is the theory around the set of properties that define vector spaces and linear mappings, and these properties are verified by a huge number of other mathematical constructions, but can also verified as first order approximations of most physical systems.

So to answer your question, if you like math for themselves, you will love Linear Algebra for its own beauty. You really start from a simple set of properties and build very interesting objects. But it is also a mandatory step for most engineering or mathematics majors, as its properties are verified by a huge number of systems.

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Well, let's try to imagine how to avoid linear algebra.

Anything that involves functions of $2$ or more variables, that can be differentiated, will very soon use linear operators or matrices. No geometry!

Also, no algebra or number theory or topology. Almost anything in algebra or its client subjects that involves associative (not necessarily commutative) multiplication, such as composition of functions, requires the simpler case of linear algebra as a preliminary.

That leaves calculus of one real or complex variable. There, it will still be hard to avoid linear algebra, unless you also avoid recursion relations, differential and difference equations, operators on the functions, measures, Fourier and Laplace transforms, special function and orthogonal polynomials, discrete or finite-element methods, calculus of variations. Sooner or later one meets linear-algebra concepts like eigenvalue, linear operator and quadratic form.

One could try to hide in parts of analysis that seem less algebraic. The Riemann zeta function does not look like it needs any linear algebra. But understanding it requires some algebraic number theory, and then there are all the conjectures about its relations to linear operators, and theorems that the zeros are spaced according to patterns typical of random... matrices. Fractal geometry doesn't look very algebraic, until you ask why the Mandelbrot set looks self-similar at all scales, or use iterated function systems, which are directly based on linear algebra. Probability theory is full of Markov chains, which are complicated linear algebra.

OK then. As long as one sticks to subjects that are outside of algebra, analysis and geometry, and are so qualitative that no serious algebra/analysis/geometry is used for calculating anything, then there might not be a need to ever know linear algebra! This excludes concrete pursuits like combinatorics, computer science and all of applied mathematics.

For the very few subjects that are left, it is still questionable whether anyone could really make use of them without knowing linear algebra. There are too many basic thought patterns that are learned in linear algebra, such as reasoning about higher-dimensional spaces, that have permeated the language and ideas in other parts of mathematics.

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Very good answers so far but I would like to add...

  • the matrix representation theory of groups (and fields) : That more advanced algebras can be calculated and systematically investigated using linear algebra of less advanced "numbers" (groups). So linear algebra for simple fields lays the foundation and provides the tools for both constructing and systematically investigating more advanced types of bananas, sorry numbers.
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Mathematical Logic, Type, Proof, and Category Theory, and Abstract Algebra do not depend on Linear Algebra that much, but rather underlay it.
Moreover say Number Theory and Homology rely on modules not vector spaces (vectors over rings, not fields).

So while Linear Algebra is indeed very widely used in practice, it is not yet any appropriate candidate for being foundation of mathematics.

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Linear algebra is often (as seems to be the case here by your instructor) taught in a disjoint way from calculus and then eventually the threads are gathered together and merged, but usually hastily with what seems to be tangential asides, leaving most students to miss the point.

The truth of the matter is it turns out that to really understand the functions you study in calculus, you need to understand spaces of functions. You already get a hint of that when you talk about differentiation and integration as being "inverse" operators. The natural structure to put on these spaces is that of a vector space (it's not coincidence that high-schoolers learn about the "algebra of functions").

Linear algebra can be considered the study of vector spaces and natural operations upon them. Using the tools of linear algebra, such as eigenvalues, you can gain an understanding of linear operators on these function spaces. This will actually help you understand differential equations, which themselves are the chief motivation for studying calculus in the first place (historically Newton et al studied calculus to solve differential equations in physics).

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