Solution Found!
Proof Let V and W be two subspaces of a vectorspace U.(a)
Chapter 4, Problem 4.3.58(choose chapter or problem)
Proof Let V and W be two subspaces of a vector space U.
(a) Prove that the set \(V+W=\{\mathbf{u}: \mathbf{u}=\mathbf{v}+\mathbf{w}, \mathbf{v} \in V \text { and } \mathbf{w} \in W\}\) is a subspace of U.
(b) Describe V + W when V and W are the subspaces of \(U=R^{2}\): V = {(x, 0): x is a real number} and W = {(0, y): y is a real number}.
Questions & Answers
QUESTION:
Proof Let V and W be two subspaces of a vector space U.
(a) Prove that the set \(V+W=\{\mathbf{u}: \mathbf{u}=\mathbf{v}+\mathbf{w}, \mathbf{v} \in V \text { and } \mathbf{w} \in W\}\) is a subspace of U.
(b) Describe V + W when V and W are the subspaces of \(U=R^{2}\): V = {(x, 0): x is a real number} and W = {(0, y): y is a real number}.
ANSWER:Step 1 of 3
Proof Let V and W be two subspaces of a vector space U.
(a) Prove that the set
\(V+W=\{\mathbf{u}: \mathbf{u}=\mathbf{v}+\mathbf{w}, \mathbf{v} \in V \text { and } \mathbf{w} \in W\}\)
is a subspace of U.
(b) Describe V + W when V and W are the subspaces of \(U=R^{2}\)
\(V=\{(x, 0): x \text { is a real number }\} \text { and } W=\{(0, y): y \text { is a real number }\} \text {. }\)