Explore BrainMass

Explore BrainMass

    Orthogonal subspaces

    Not what you're looking for? Search our solutions OR ask your own Custom question.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    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 December 24, 2021, 5:05 pm ad1c9bdddf

    Solution Summary

    There are several proofs here regarding matrices, including proving same rank, nonsingularity, and subspaces.