Let A be an n x n matrix. Denote it's columns b v_{1}, v_{2}, ... v_{n}. Keep in mind these are column vectors in R^{n}.
Prove that A is invertible if and only if {v_{1}, v_{2}, ... v_{n}} spans R^{n}.
I have no idea what you know and can use. But here is a quick proof using things I feel ought to be done before proving this theorem.
spans
iff
is consistent for all
iff
has pivot rows (and also pivot columns since is square)
iff
is row equivalent to the identity matrix
iff
is invertible.
First show that if you take A times you get the first column of A, that if you take A times you get the second column, etc.
In other words, A maps the standard basis for R^{n} into the columns of A. Those vectors span the range of A.