91Ó°ÊÓ

Let \(A\) be an \(n \times n\) matrix. Prove that if \(A\) is nonsingular and \(\left\\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{k}\right\\}\) is linearly independent, then \(\left\\{A \mathbf{v}_{1}, A \mathbf{v}_{2}, \ldots, A \mathbf{v}_{k}\right\\}\) is likewise linearly independent. Give an example to show that the result is false if \(A\) is singular.

Let \(U\) and \(V\) be subspaces of \(\mathbb{R}^{n}\). Prove that \(\operatorname{dim}(U+V)=\operatorname{dim} U+\operatorname{dim} V-\) \(\operatorname{dim}(U \cap V)\). (Hint: This is a generalization of Exercise 20. Start with a basis for \(U \cap V\), and use Exercise 17.)

a. Let \(A\) be an \(m \times n\) matrix, and let \(B\) be an \(n \times p\) matrix. Show that \(A B=\mathrm{O} \Longleftrightarrow\) \(\mathbf{C}(B) \subset \mathbf{N}(A)\). b. Suppose \(A\) and \(B\) are \(3 \times 3\) matrices of rank 2 . Show that \(A B \neq \mathrm{O}\). c. Give examples of \(3 \times 3\) matrices \(A\) and \(B\) of rank 2 so that \(A B\) has each possible rank.