Chapter 6, Problem 13SE

Use the steps below to prove the following relations among the four fundamental subspaces determined by an m × n matrix A.

Row A = (Nul A)^⊥, Col A = (Nul A^T)^⊥

1. a. Show that Row A is contained in (Nul A)^⊥. (Show that if x is in Row A, then x is orthogonal to every u in Nul A.)
2. b. Suppose rank A = r. Find dim Nul A and dim (Nul A)^⊥, and then deduce from part (a) that Row A = (Nul A)^⊥. [Hint: Study the exercises for Section 6.3.]
3. c. Explain why Col A = (Nul A^T)^⊥.