Hwk_8_soln (4.1)

Hwk_8_soln (4.1) - M3710 Linear Algebra Instructor Phoebe...

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Unformatted text preview: M3710: Linear Algebra Instructor: Phoebe McLaughlin Fall 2011 Homework 8 Solution (4.1) 4.1 Vector Spaces and Subspaces Selected assignment: 6, 8, 10, 14, 16, 18, 26, 32 6. The zero polynomial cannot be written in the format of That is, the zero polynomial is not in the set subspace of . 8. Let . Prove that is a subspace of Proof. 1) Consider the zero polynomial , since 2) For any and . Since , For any , , . Hence is a subspace of . 14. for any . Hence . . . . is not a M3710: Linear Algebra Instructor: Phoebe McLaughlin Fall 2011 Homework 8 Solution (4.1) 32a. Let and be subspaces of a vector space . Prove that is a subspace of . Proof. 1) Both and contain the zero vector of because they are subspaces of . Thus . 2) For any and , respectively. Since is a subspace of , . Likewise and is a subspace of . Thus . For any , and , since are subspaces of . Thus . Therefore, is a subspace of . 32b. Give an example in subspace. to show that the union of two subspaces is not, in general, a Example: Let Then But Thus and . Then both and and are subspaces of . . That is, is not a subspace of . is not closed under vector addition. . M3710: Linear Algebra Instructor: Phoebe McLaughlin Fall 2011 Homework 8 Solution (4.1) ...
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