Does row independence imply column independence?
The first way to prove it that I can think of is the following:
- A matrix has non-zero determinant if and only if it's rows are independent;
- A matrix has non-zero determinant if and only if the transposed matrix does.
So, start from a matrix with independent rows. So, its determinant is non-zero. Then also the transposed matrix has a non-zero determinant. This implies that the rows of the transposed matrix are independent too, but the rows of the latter are the column of the first.