Define Z as an m×n matrix. Show that Z has rank m if and only if the determinant of some m × m sub-matrix of Z, which is obtained by deleting the n − m columns of Z, is nonzero.
I have managed to show the forward direction for this proof. I'm stuck in showing the reverse however (Starting with the submatrix and the determinant of the submatrix being non-zero). I know that it has m linear independent columns, thus rank = m. How can I always know that adding the n-m columns back keeps the rank = m. Must I just assume they're linearly dependent from the other m columns? Thank you.