91Ó°ÊÓ

Let \(A\) and \(B\) be \(n \times n\) matrices. (a) Show that \(A B=O\) if and only if the column space of \(B\) is a subspace of the null space of \(A\) (b) Show that if \(A B=O\), then the sum of the ranks of \(A\) and \(B\) cannot exceed \(n\).

Let \(S, T,\) and \(U\) be subspaces of a vector space \(V .\) We can form new subspaces using the operations of \(\cap\) and \(+\) defined in Exercises 23 and 25 When we do arithmetic with numbers, we know that the operation of multiplication distributes over the operation of addition in the sense that \\[ a(b+c)=a b+a c \\] It is natural to ask whether similar distributive laws hold for the two operations with subspaces. (a) Does the intersection operation for subspaces distribute over the addition operation? That is does \\[ S \cap(T+U)=(S \cap T)+(S \cap U) \\] (b) Does the addition operation for subspaces distribute over the intersection operation? That is does \\[ S+(T \cap U)=(S+T) \cap(S+U) \\]

Let \(A \in \mathbb{R}^{m \times n}, B \in \mathbb{R}^{n \times r},\) and \(C=A B .\) Show that (a) the column space of \(C\) is a subspace of the column space of \(A\) (b) the row space of \(C\) is a subspace of the row space of \(B\) (c) \(\operatorname{rank}(C) \leq \min \\{\operatorname{rank}(A), \operatorname{rank}(B)\\}\).

An \(m \times n\) matrix \(A\) is said to have a right inverse if there exists an \(n \times m\) matrix \(C\) such that \(A C=I_{m}\) The matrix \(A\) is said to have a left inverse if there exists an \(n \times m\) matrix \(D\) such that \(D A=I_{n}\) (a) Show that if \(A\) has a right inverse, then the column vectors of \(A\) span \(\mathbb{R}^{m}\) (b) Is it possible for an \(m \times n\) matrix to have a right inverse if \(n