Assume that A inR^(nxn) and without using the invertible matrix theorem, prove the following:
3.1.Spanning Sets. If A is an n x n matrix and A^(-1) exists, then the columns of A span R^n.
3.2. Pivot Structure. If A is an n x n matrix and Ax = b has a solution for each b inR^n, then A is invertible.
3.3. Linear Independence. If the matrix A is invertible, then the columns of A^(1)are linearly independent.
I'm not sure how to do any of these. I don't know where to begin or anything. If someone could guide me through them that would be very helpful. Thank you