In one of his online lectures Benedict Gross comments that one can never have too much Linear Algebra. Also, looking around it seems like I can find comments to the effect that Linear Algebra has more importance than other sub-disciplines of mathematics. If true, why is Linear Algebra the most important sub-discipline of mathematics? If Benedict Gross is correct, why can't one have enough Linear Algebra?
This is a bit of a broad answer, so don't be too critical.
In all of mathematics, the idea of "linearization" is quite an important one. Linear problem are in some sense very well understood, and we can say basically everything we want about them. Because of this, approaches to many other areas consist of making things linear so we can solve it.
So why learn linear algebra? Because it is the most well understood, and is used to understand more difficult problems in mathematics.
Numerical methods for ODE's, and dynamical systems.
A major aspect of Representation theory is a way to learn more by "linearizing" groups, and using linear algebra to solve otherwise difficult group theoretic problems.
The derivative in calculus is simply the linear approximation to a function at point.
In graph theory, to say things about a random walk, it easiest to make the graph into a vector space and consider random-walk operators on that space.
Hope that helps,
ADDED: Since there are certainly many more examples, I made this post community wiki so others can add to this list.
I agree. Linear algebra is one of the most important branches of pure mathematics. It is used in the following areas of pure mathematics, for example:
Representation Theory (indeed, the basic purpose of representation theory is to solve problems in algebra by transferring them to the linear algebra context)
Module Theory (module theory is in some sense a generalization of linear algebra: it is linear algebra done over arbitrary rings rather than over fields)
Funtional Analysis (indeed, functional analysis is, in some sense, infinite-dimensional linear algebra)
ODE's and PDE's
Differential Geometry (indeed, a main purpose of differential geometry is to understand maps between manifolds by approximating them by linear maps)
In fact, it is hard to think of a branch of mathematics where linear algebra is not used. I could continue the list above and include more and more branches of mathematics but this would take too long.
The ubiquity of vector spaces in mathematics is indisputable. The abstractions created in linear algebra rest on a huge collection of incredibly useful examples.
These abstractions have multifarious applications (computer graphics, optimization, dynamical systems, etc). The abstraction of vector space is accessible and extremely useful.
Linear algebra unifies many fields of inquiry in an elegant and efficient way.