LetAbe annnmatrix. Prove thatAis nonsingular if and only if the columns ofAspanF^n. You may use the most recent version of the nonsingularity theorem in your proof.

Nonsingularity theorem:

LetAbe annnmatrix. The following are equivalent.

• Ais nonsingular.
• ThenullspaceofAisN(A)={0}.
• The row reduced echelon form ofAisIn.
• TherankofAisn.
• The homogeneous system of linear equations with coefficient matrixAhas only the trivial solution.
• The system of linear equations with coefficient matrixAand constant vectorbhas a unique solution for any choice ofb.
• Ais invertible.
• The columns ofAare linearly independent.
• The columns ofAform a basis ofFn.

This is all the information I've been given!!!!