Let A be an mxn matrix. show that
1) If x Є N(A^TA), then Ax is in both R(A) and N(A^T).
2) N(A^TA) = N(A.)
3) A and A^TA have the same rank.
4) If A has linearly independent columns, then A^TA is nonsingular.
Let A be an mxn matrix, B an nxr matrix, and C=AB. Show that:
1) N(B) is a subspace of N(C).
2) N(C) perp. is a subspace of N(B) perp. and consequently, R(C^T) is a subspace of R(B^T).© BrainMass Inc. brainmass.com March 4, 2021, 6:03 pm ad1c9bdddf
There are several proofs here regarding matrices, including proving same rank, nonsingularity, and subspaces.