MA353_S10_mid1_practice_sol

MA353_S10_mid1_practice_sol - MA 353, Practice Midterm 1...

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Unformatted text preview: MA 353, Practice Midterm 1 Feb 17, 2010 Name: This exam consists of 7 pages including this front page. Ground Rules (1) No calculator is allowed. (2) Show your work for every problem unless stated otherwise. A correct an- swer without any justification may not receive the full credit. (3) You may use one 4-by-6 index card, both sides. (4) Show your ID on your desk. Score 1 10 2 10 3 10 4 10 5 10 Total 50 1 2 1. Prove or disprove the following sets are subspaces. (a) V = { ( x 1 ,x 2 ,x 3 ) R 3 : x 3 = x 1 + x 2 } . Answer This is a subspace. Proof. Let ( x 1 ,x 2 ,x 3 ) and ( y 1 ,y 2 ,y 3 ) be in V . Then x 3 = x 1 + x 2 and y 3 = y 1 + y 2 . Now ( x 1 ,x 2 ,x 3 ) + ( y 1 ,y 2 ,y 3 ) = ( x 1 + y 1 ,x 2 + y 2 ,x 3 + y 3 ) , and x 3 + y 3 = ( x 1 + x 2 ) + ( y 1 + y 2 ) = ( x 1 + y 1 ) + ( x 2 + y 2 ) , which shows that V is closed under addition. Next let c R . Then c ( x 1 ,x 2 ,x 3 ) = ( cx 1 ,cx 2 ,cx 3 ) , and cx 3 = c ( x 1 + x 2 ) = cx 1 + cx 2 , which shows that V is closed under scalar multiplication.is closed under scalar multiplication....
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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 University-West Lafayette.

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MA353_S10_mid1_practice_sol - MA 353, Practice Midterm 1...

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