Let A be an n times n matrix If the columns of A are linearl

Let A be an n times n matrix. If the columns of A are linearly independent, then are the rows of A linearly independent? Explain your answer.

Solution

Solition: Linearly independent rows of A are linearly independent columns of A^T, and linearly independent columns of A^T make A^T invertible, which in turn makes A invertible, which finally gives linearly independent columns of A. The reverse is also true. If the rows of A are linearly independent, then the result of doing row-reduction to A is the identity matrix, so the only solution of Av = 0 is v = 0.

So, if the columns are linearly independent, then are the rows of A linearly independent.
.