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(10 points) Suppose V is & vector space and U and W are subspaces of V Use Theorem TSS on page 274 of Beezer to prove that UnW is a subspace of V . [See the doc...

Question

(10 points) Suppose V is & vector space and U and W are subspaces of V Use Theorem TSS on page 274 of Beezer to prove that UnW is a subspace of V . [See the document titled "A brief introduction to sets" in the Course Materials folder on Blackboard for the definition of the intersection of two sets:]

(10 points) Suppose V is & vector space and U and W are subspaces of V Use Theorem TSS on page 274 of Beezer to prove that UnW is a subspace of V . [See the document titled "A brief introduction to sets" in the Course Materials folder on Blackboard for the definition of the intersection of two sets:]



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Let $S_{1}$ and $S_{2}$ be subspaces of a vector space $V .$ Let $$ \begin{array}{l} S_{1} \cup S_{2}=\left\{\mathbf{v} \in V: \mathbf{v} \in S_{1} \text { or } \mathbf{v} \in S_{2}\right\} \\ S_{1} \cap S_{2}=\left\{\mathbf{v} \in V: \mathbf{v} \in S_{1} \text { and } \mathbf{v} \in S_{2}\right\} \end{array} $$ and let $$S_{1}+S_{2}=\left\{\mathbf{v} \in V: \mathbf{v}=\mathbf{x}+\mathbf{y} \text { for some } \mathbf{x} \in S_{1} \text { and } \mathbf{y} \in S_{2}\right\}$$ (a) Show that, in general, $S_{1} \cup S_{2}$ is not a subspace of $V$ (b) Show that $S_{1} \cap S_{2}$ is a subspace of $V$. (c) Show that $S_{1}+S_{2}$ is a subspace of $V$.


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