Why study linear algebra? Simply as the title says. I've done some research, but still haven't arrived at an answer I am satisfied with. I know the answer varies in different fields, but in general, why would someone study linear algebra?
Linear algebra is vital in multiple areas of science in general. Because linear equations are so easy to solve, practically every area of modern science contains models where equations are approximated by linear equations (using Taylor expansion arguments) and solving for the system helps the theory develop. Beginning to make a list wouldn't even be relevant ; you and I have no idea how people abuse of the power of linear algebra to approximate solutions to equations. Since in most cases, solving equations is a synonym of solving a practical problem, this can be VERY useful. Just for this reason, linear algebra has a reason to exist, and it is enough reason for any scientific to know linear algebra.

More specifically, in mathematics, linear algebra has, of course, its use in abstract algebra ; vector spaces arise in many different areas of algebra such as group theory, ring theory, module theory, representation theory, Galois theory, and much more. Understanding the tools of linear algebra gives one the ability to understand those theories better, and some theorems of linear algebra require also an understanding of those theories ; they are linked in many different intrinsic ways.

Outside of algebra, a big part of analysis, called functional analysis, is actually the infinite-dimensional version of linear algebra. In infinite dimension, most of the finite-dimension theorems break down in a very interesting way ; some of our intuition is preserved, but most of it breaks down. Of course, none of the algebraic intuition goes away, but most of the analytic part does ; closed balls are never compact, norms are not always equivalent, and the structure of the space changes a lot depending on the norm you use. Hence even for someone studying analysis, understanding linear algebra is vital.

In other words, if you wanna start thinking, learn how to think straight (linear) first. =)

Hope that helps,
I was browsing the posts flaired Mathematics at r/askscience, and lighted upon the same question!

I'll replicate simply the top-voted post and its top-voted follow-up comment. 



AirborneRodent

Let me give a concrete example.  I use linear algebra every day for my job, which entails using finite element analysis for engineering.

Imagine a beam.  Just an I-beam, anchored at one end and jutting out into space.  How will it respond if you put a force at the end?  What will be the stresses inside the beam, and how far will it deflect from its original shape?

Easy.  We have equations for that.  A straight, simple I-beam is trivial to compute.

But now, what if you don't have a straight, simple I-beam? What if your I-beam juts out from its anchor, curves left, then curves back right and forms an S-shape?  How would that respond to a force?  Well, we don't have an equation for that.  I mean, we could, if some graduate student wanted to spend years analyzing the behavior of S-curved I-beams and condensing that behavior into an equation.

We have something better instead: linear algebra.  We have equations for a straight beam, not an S-curved beam.  So we slice that one S-curved beam into 1000 straight beams strung together end-to-end, 1000 finite elements.  So beam 1 is anchored to the ground, and juts forward 1/1000th of the total length until it meets beam 2.  Beam 2 hangs between beam 1 and beam 3, beam 3 hangs between beam 2 and beam 4, and so on and so on.  Each one of these 1000 tiny beams is a straight I-beam, so each can be solved using the simple, easy equations from above.  And how do you solve 1000 simultaneous equations?  Linear algebra, of course!



MiffedMouse

And to be clear, this kind of situation shows up everywhere.

Atomic orbitals? Check

Fluid flow? Check

Antenna radiation patterns? Check

Face recognition? Check

Honestly, anything that involves more than one simple element probably uses linear algebra.