Let
A ∈ Mn(F).
Prove that A ∈ GLn(F) if and only if the columns of A form a basis for Fn.
$\displaystyle GL_n(F)$ is the group of invertible n by n matrices on F. The columns of a matrix span the image of the matrix in $\displaystyle F^n$. If those columns form a basis for $\displaystyle F^n$, then the image is the entire space and so the matrix is invertible.