Take your garden variety high-school algebra, but use matrices instead of scalars. Lets you do fun stuff like eigenvalues, eigenvectors, linear transformations, vectors, Markov chains. Very useful in physics, statistics, and multivariable calculus, or other mathematically tasks that require processing a large bunch of numbers in similar ways.

In addition to being an interesting area of mathematics, linear algebra is also, at least in the United States, one of the first courses that deals with abstract thought and rigorous proof. In most of the course descriptions I have read, linear algebra usually covers systems of linear equations, matrix algebra (matrix addition, matrix multiplication, elementary row operations, row reduction, etc), linear independence, inverses, the determinant, Cramer's rule, linear transformations, subspaces, vector spaces, eigenvalues, eigenvectors, and diagonalization.

Since I'm just a student, I won't make conjectures as to courses outside of the one I am taking. My instructor basically has commited the course to pre-abstract algebra, so it's probably more proof-intensive than the average sophomore-level course. However, proof and abstraction is unavoidable when dealing with vector spaces and subspaces. The hardest part I had with the latter proofs (the incredibly simple "prove that a subspace H of a vector space V is also a vector space," for example) was letting go of Rn. For two years, including high school, everything that I've done has been with the reals and with vectors in R2 and R3 in multi-variable calculus. Having to give up my precious Rn for the arbitrary V was very hard, but once I did, the proofs just fell into place. Everywhere in calculus we always had graphs with nice numbers and fairly simple geometric representations, but with arbitrary vector spaces, these tools fail in many places. Since those tools fail, new tools must be built. Although I know I will have much more trouble with next semester's group theory course, I can't help but feel that linear algebra has been a good elementary tool-building course.

Needless to say, if you're at all interested in mathematics, I'd recommend you take a linear algebra course.

In addition to the applications that soren.harward mentioned, linear algebra, together with trigonometry are the key areas of mathematics you have to have a solid knowledge of to be a computer games developer. The game world and its objects in modern 3D games are made out of triangles, which are made of 3 vertices (points in 3D space) each.
Vertices are 3- or 4-dimensional vectors (x,y,z,w), while things such as object movement, camera rotations, and the perspective projection that brings the 3D world to your 2D computer screen are matrices. So basically, the GPU on a modern graphics card is a very fast linear algebra calculator that does nothing but billions of vector/matrix operations per second.

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