115hw1solutions - Math 115A Homework 1 Due October 10th,...

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Unformatted text preview: Math 115A Homework 1 Due October 10th, 2008 1. For each of the following, check if W V is an F-subspace of the F-vector space V . If yes, write out a proof. If not, prove that. (2 pts each) a) F = R , V = R 3 and W = { v = ( v 1 ,v 2 ,v 3 ) V | 2 v 1- v 2 = 1 } . Since 2 - 6 = 1, (0 , , 0) / W , so W is not a subspace. b) F = Q , V = R and W = { x R | x 2 Q } . Take x W and y W , and let Q . Then 2( x + y ) = 2 x + 2 y Q , so x + y W . Moreover, 2( x ) = ( 2 x ) Q , so that x W . Finally, 2 0 = 0 Q , whence 0 W . That is, W is a subspace. c) F = R , V = C ( R ) the set of continuous real-valued functions on the real numbers and W = { f V | R 1 f ( x ) dx = 0 } . Let f W and g W , and let R . Then R 1 ( f + g )( x ) dx = R 1 f ( x ) dx + R 1 g ( x ) dx = 0 + 0 = 0, so that f + g W . Also, R 1 ( f )( x ) dx = R 1 f ( x ) dx = 0 = 0, so f W . Finally, 0 is obviously in W . Therefore, W is a subspace. 2. For each of the following subsets S V of the F-vector space V , check if S is linearly independent. Prove you assertions. (2 pts each) a) F = Q , V = R and S = { 1 , 2 } ....
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This note was uploaded on 02/21/2010 for the course MATH 115A 262398211 taught by Professor Fuckhead during the Spring '10 term at UCLA.

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115hw1solutions - Math 115A Homework 1 Due October 10th,...

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