EasyOrthogonals

In this caption I write B for the transpose of A.

We saw how to find nice bases for R(A) and R(B) with a simultaneous forwards sweep of Ax=y and By=x. Solving the consistency conditions you find vectors such as the ones on the board.

Then you find the null spaces N(B) and N(A) as shown.

NB The v on the third line should have subscript 4.

Theorem 3 in Lay's Section 6.1 should be read as a statement about R(B) and N(A), and a statement about R(A) and N(B).

Then note that

R(A) is the column space of A.
R(B) the row space of A.