A matrix is just a list of where a linear map sends each basis element (the nth column of a matrix is the output vector for the nth input basis vector). Lots of things are linear (e.g. scaling, rotating, differentiating, integrating, projecting, and any weighted sums of these things). Lots of other things are approximately linear locally (the derivative if it exists is the best linear approximation. i.e. the best matrix to approximate a more general function), and e.g. knowing the linear behavior near a fixed point can tell you a lot about even nonlinear systems.
Translation to something ordinary humans can understand (I personally have strong distaste for the basis vector/basis element approach):
Linear transformations 101:
If the scalar function f(x)=y is linear, what can it do?
We can now either pretend that canonical forms do not exist, or we just use them anyway and make some people angry. The former way is very long and I don't have much time so angry people are a small price to pay.
There is one way to describe f and that is as f(x)=ax=y.
Now what if we have multiple inputs f(a,b)=y? We can just have two parameters. f(a,b) = c*a+d*b = y. I can already hear the boos.
Note that c*a+d*b has the same form as a dot product. A linear function with multiple inputs and multiple outputs can be described as: f(x) = a^Tx = y, where x is a vector of dimension 2 containing the original a and b, and a is a vector containing c and d. Multiplication with a row vector is a linear transformation.
Now what if we do the opposite? One input, multiple outputs? It's the same but with column vectors! f(x) = xa = y where a is a vector and y is a vector.
Now what if we do both? Multiple inputs and multiple outputs?? You just stack either your row vectors or the column vectors! Now you have a matrix as your parameter for f(x)=Ax=y!
The best part? I've already given four examples of linear transformations including the dual space. Having canonical representations makes explanations very easy.
Yes, I think of them as saying "and this is what the coordinates in our coordinate system [basis] shall mean from now on". Systems of nonlinear equations, on the other hand, are some kind of sea monsters.
