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Unformatted text preview: Purdue University MA 353: Linear Algebra II with Applications Homework 2, due Jan. 29, solutions (I) Sec. 1.3 #19: Let W 1 and W 2 be subspaces of a vector space V . Prove that W 1 S W 2 is a subspace of V if and only if W 1 ⊆ W 2 or W 2 ⊆ W 1 . Proof. (= ⇒ ) Assume W 1 S W 2 is a subspace of V . We must show W 1 ⊆ W 2 or W 2 ⊆ W 1 . We will prove it by contradiction. So assume the contrary, ie W 1 * W 2 and W 2 * W 1 . Then there exit w 1 ∈ W 1 and w 2 ∈ W 2 such that w 1 / ∈ W 2 and w 2 / ∈ W 1 . Now consider w 1 + w 2 . Since W 1 S W 2 is a subspace of V by our assumption, we have w 1 + w 2 ∈ W 1 [ W 2 . Therefore w 1 + w 2 ∈ W 1 or w 1 + w 2 ∈ W 2 . Now let w = w 1 + w 2 . If w 1 + w 2 ∈ W 1 , then w ∈ W 1 . But w 2 = w w 1 and so if w ∈ W 1 , then w w 1 ∈ W 1 because W 1 is a subspace. Hence w 2 ∈ W 1 . This contradicts to the assumption that w 2 / ∈ W 1 ....
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This note was uploaded on 04/29/2010 for the course MA 353 taught by Professor Staff during the Spring '08 term at Purdue.
 Spring '08
 Staff
 Linear Algebra, Algebra, Vector Space

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