[Brother Daniel]

Why "linear"?

Good (and well timed) question.

My description was perhaps too broad, since there are things you can do with vector spaces that go outside the scope of Linear Algebra.

But suppose the only functions of vectors that we consider are those having the properties
f(u+v) = f(u) + f(v) for any vectors u and v;
f(cv) = cf(v) for any vector v and any scalar c.
(This is what we call a "linear function".)

That's a rather restricted set of functions, but it turns out that there's a tremendous amount of stuff to learn about them, so they're worthy of having entire courses dedicated to them. That's Linear Algebra.