A is invertible iff the columns (and rows) of A are linearly independent

A is invertible

iff det |A| is non-zero

iff rank A is n

iff column rank is n

iff dim (column space is n)

iff the n columns of A are linearly independent

Well, this is a proof that I laid down. It was junked by my prof. She said I have to use linear transformations to prove it. Someone throw some light ?