91Ó°ÊÓ

In \(\mathbb{R}^{4}\), let \(U\) be the subspace of all vectors of the form \(\left(u_{1}, u_{2}, 0,0\right)^{T},\) and let \(V\) be the subspace of all vectors of the form \(\left(0, v_{2}, v_{3}, 0\right)^{T}\). What are the dimensions of \(U, V, U \cap V, U+V ?\) Find a basis for each of these four subspaces. (See Exercises 20 and \(22 \text { of Section } 2 .)\)

Let \(S\) be the vector space of infinite sequences defined in Exercise 15 of Section \(1 .\) Let \(S_{0}\) be the set of \(\left\\{a_{n}\right\\}\) with the property that \(a_{n} \rightarrow 0\) as \(n \rightarrow \infty\) Show that \(S_{0}\) is a subspace of \(S\).

Show that if \(U\) and \(V\) are subspaces of \(\mathbb{R}^{n}\) and \(U \cap V=\\{0\\},\) then \\[ \operatorname{dim}(U+V)=\operatorname{dim} U+\operatorname{dim} V \\]

Let \(U\) and \(V\) be subspaces of a vector space \(W\) Prove that their intersection \(U \cap V\) is also a subspace of \(W\)

Let \(S, T,\) and \(U\) be subspaces of a vector space \(V\) We can form new subspaces by using the operations of \(\cap\) and \(+\) defined in Exercises 20 and \(22 .\) 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 disdoes tribute over the intersection operation? That is, \\[ 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) if \(A\) and \(B\) both have linearly independent column vectors, then the column vectors of \(C\) will also be linearly independent. (b) if \(A\) and \(B\) both have linearly independent row vectors, then the row vectors of \(C\) will also be linearly independent. [Hint: Apply part (a) to \(C^{T}\).]