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# hw1sol - which is linearly dependant ∅ ⊂ is linearly...

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UCB Math 110, Spring 2011: Homework 1 Solutions to Graded Problems 1.2.1(b) False. Let 0 , 0 0 V be two zero vectors. Then 0 = 0 + 0 0 = 0 0 . 1.3.23(a) First we show that W 1 + W 2 is a subspace. As both W i is a subspace of V for all i , we have 0 W i . Thus 0 + 0 = 0 W 1 + W 2 . Let x, y W 1 + W 2 and c F . Then x = x 1 + x 2 and y = y 1 + y 2 for some x i , y i W i for all i . A simple calculation gives that x + cy = ( x 1 + cy 1 ) + ( x 2 + cy 2 ) . Once again W i is a subspace of V for all i giving that x i + cy i W i . Thus x + cy W 1 + W 2 , implying that W 1 + W 2 is a subspace of V . Let x W 1 . As 0 W 2 , we get that x + 0 = x W 1 + W 2 . Hence W 1 W 1 + W 2 . Similarly W 2 W 1 + W 2 . 1.5.1(d) False. Counter Example: Consider the set {
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Unformatted text preview: which is linearly dependant. ∅ ⊂ { } is linearly independant. 1.6.22 The condition is that W 1 ⊂ W 2 , thus we must prove that ”dim( W 1 ∩ W 2 ) = dim( W 1 ) if and only if W 1 ⊂ W 2 .” Let dim( W 1 ∩ W 2 ) = dim( W 1 ). As W 1 ∩ W 2 ⊂ W 1 , we see that our assumption gives that W 1 ∩ W 2 = W 1 . Thus W 1 = W 1 ∩ W 2 ⊂ W 2 . Conversely let W 1 ⊂ W 2 . Thus W 1 ∩ W 2 = W 1 giving that dim( W 1 ∩ W 2 ) = dim( W 1 )....
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