Comment by ljlolel

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.