homework set 12-Solutions - B ˙ ILKENT UNIVERSITY...

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Unformatted text preview: B ˙ ILKENT UNIVERSITY Department of Mathematics MATH 225, LINEAR ALGEBRA and DIFFERENTIAL EQUATIONS, Solution of Homework set 1 # 12 U. Mu˘gan July 2, 2008 Homework problems from the 2 nd Edition, SECTION 4.2 3 (3) ) 2 A typical vector in W is of the form ~x = ( x 1 , 1 ,x 3 ). Since scalar multiple c~x = ( cx 1 ,c,cx 3 ) is not in W unless c = 1. Hence W is not closed under multiplication by scalar, and therefore W is not a subspace of R 3 . 7 (7) ) Note that the vectors ~x = (1 , 1) and ~ y = (1 ,- 1) are in W . But their sum ~x + ~ y = (2 , 0) is not in W , because | 2 | 6 = | | . Hence, W is not a subspace of R 2 . 8 (8) ) Since x 2 1 + x 2 2 = 0, W is simply the zero subspace { ~ } of R 2 . 10 (10) ) The vectors ~x = (1 , 0) and ~ y = (0 , 1) are in W . But their sum ~x + ~ y = (1 , 1) is not in W , because | 1 | + | 1 | = 2 6 = 1 . Hence, W is not a subspace of R 2 . 11 (11) ) Let ~x = ( x 1 ,x 2 ,x 3 ,x 4 ) and ~ y = ( y 1 ,y 2 ,y 3 ,y 4 ) be the vectors in W , so x 1 + x 2 = x...
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This note was uploaded on 06/18/2010 for the course COMPUTER S Math 225 taught by Professor Yosumhoca during the Spring '10 term at Bilkent University.

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homework set 12-Solutions - B ˙ ILKENT UNIVERSITY...

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