math110s-hw2 - 2 for the direct sum of three or more...

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MATH 110: LINEAR ALGEBRA SPRING 2007/08 PROBLEM SET 2 1. Let V be a vector space over F . Let w V be a fixed non-zero vector and μ F be a fixed non-zero scalar. (a) Show that the function f : F V defined by f ( λ ) = λ w is injective. (b) Show that the function g : V V defined by g ( v ) = μ v is bijective. (c) Show that the function h : V V defined by h ( v ) = v + w is bijective. 2. Let W 1 and W 2 be subspaces of a vector space V . The sum of W 1 and W 2 is the subset of V defined by W 1 + W 2 = { w 1 + w 2 V | w 1 W 1 , w 2 W 2 } . (a) Prove that W 1 + W 2 is a subspace of V . (b) Prove that W 1 + W 2 is the smallest subspace of V containing both W 1 and W 2 . (c) Prove that W 1 W 2 is the largest subspace of V contained in both W 1 and W 2 . 3. Let W 1 and W 2 be subspaces of a vector space V . Show that the following statements are equivalent. (i) W 1 W 2 = { 0 } . (ii) If w 1 W 1 and w 2 W 2 are such that w 1 + w 2 = 0 , then w 1 = w 2 = 0 . (iii) If w 1 + w 2 = w 0 1 + w 0 2 , where w 1 , w 0 1 W 1 and w 2 , w 0 2 W 2 , then w 1 = w 0 1 and w 2 = w 0 2 . If any one of these equivalent conditions holds, then W 1 + W 2 is written W 1 W 2 and is called the direct sum of W 1 and W 2 . 4. (a) State and prove the analogue of the statements in Problem
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Unformatted text preview: 2 for the direct sum of three or more subspaces. (b) Let W 1 ,W 2 ,W 3 be subspaces of a vector space V . Suppose W 1 W 2 = W 1 W 3 = W 2 W 3 = { } . Must W 1 + W 2 + W 3 be a direct sum? 5. Prove or provide a counter example for the following. (a) Let V 1 := a b-b a R 2 2 a,b R , V 2 := c d d-c R 2 2 c,d R . Is it true that R 2 2 = V 1 V 2 ? (b) Let W 1 := { p ( x ) P 3 | p (-x ) = p ( x ) for all x R } , W 2 := { p ( x ) P 3 | p (-x ) =-p ( x ) for all x R } . Is it true that P 3 = W 1 W 2 ? Date : February 14, 2008 (Version 1.0); due: February 21, 2008. 1...
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