Linear Algebra is a little bit of a funny subject-- virtually every linear algebra problem will encounter at least one matrix, but its really no more about matrices than literature is about paper and ink.
The other day I was thinking about my linear algebra class from a few years ago, and realized that, while I could discuss most of the important points coherently, I could not remember the first thing about how it related to a matrix.
I think it's worthwhile to condense a subject to its central insight, so my best attempt for linear algebra is this: "one vector defines a set of all vectors that are just like it, but bigger or smaller, and this set of vectors is a line passing through the origin. If you add in a second vector, that is not on this line, you have defined a plane, because all points on this plane can be described as some combination of the first two vectors."
To be a bit more concrete about this, imagine a town neatly laid out in a grid pattern (not needed, but conceptually useful), and imagine town hall sitting at the center of town. Now imagine I give you the instruction "go three blocks north." This is a vector--I've given a distance and a direction. I've also defined a set of instructions though--the instructions "go 6 blocks north" or "go 4 blocks south" are both closely related to the first direction, because you could re-write them as "do the first instruction twice" or "do the opposite of the first instruction one and a third times." This are linear transformations of the first instruction. Now, if I give you a completely different instruction, say "go one block west," I have described a new line, in the East-West direction. The really cool thing though is that we can take those two instructions ("go 3 blocks north" and "go 1 block west") and put them together to describe any point on the map. We needed two separate vectors to describe a two dimensional space, and we'd need three separate vectors to describe a three dimensional space, and so on, but once we have these vectors (a "basis set"), we can add them together in different ways to describe the entire space.