Say you start with a small system of equations and you find out how you can solve them. Then you solve a bunch more, but then you start to notice that you don't

Say you start with a small system of equations and you find out how you can solve them. Then you solve a bunch more, but then you start to notice that you don't like writing out the x, y, and z's, so you just keep the coefficients, in a box of sorts. Then you translate what you were doing into operations on the rows of this ... matrix you call it. After solving some more equations, you start to forget where these things come from. Surely they come from equations, but because the coefficients, plus signs and equal signs disappear, you are left with a box of numbers really. What happens next is that different questions start to come to mind. The matrix attains a life of its own. You start to think questions like "Well, we have some x, y, z's and this box of numbers somehow transforms them into the right hand side's a, b, c's". Ok, the matrix is some kind of "transformation" now. You look at its properties. It appears that aX + bY can be computed entry-wise, so maybe this is a fundamental property and you call it linearity. Hmm. At some point you forget about the boxes of numbers and now you have bonafide transformations. From what? To where? You call these...vector spaces. And so on. Further down the line you become preoccupied by questions of a different flavour, like what happens if you want something like Ax = cx. That is, we get x back with minimal distortion. Fairly simple question that leads to a lot of math. It also just happens that when you consider mathematical functions of several dimensions, and matrices become apparent when you consider linear approximations. You've built this great theory for them and now that theory seems to really help in these kinds of problems. That is, a LOT of computational problems in this world.