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Orthogonal subspaces

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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).

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Solution Summary

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

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