Suppose we have an nxn matrix A with column vectors v1,...,vn. A is invertible. With rank(A)=n. How do I prove that v1,...,vn are linearly independent? I think I can prove this by using the fact that rank(A)=n, which tells me that there is a pivot in each of the n columns of the rref(A) matrix (because rref(invertible mx) gives an identity matrix). I'm not sure how to interpret this result to show that each column vector are linearly independent though. Should I look at the linear combination of an identity matrix to establish independence?