The vectors of f span the image space iff f is onto.
If f is 1-1 and onto, the domain and range have the same dimension, hence the matrix is square, and nonsingular. Conversely, a nonsingular matrix represents a 1-1 function from n space onto itself.
If f is 1-1 and onto, let g be the inverse map. Thus f(g(y)) = y. If g(y) = x, then f(g(ky) = ky, which forces g(ky) = kx. A similar argument shows g commutes with addition, hence g is a linear function. Its nonsingular matrix is the inverse of the matrix associated with f.
The set of invertible linear functions from a vector space onto itself forms a group. This is the same as the group of nonsingular matrices. Many interesting subgroups are possible, such as the group of rigid rotations about the origin.