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Unformatted text preview: Math 235 Sample Term Test 1  2 Answers NOTE :  Only answers are provided here (and some proofs). On the test you must provide full and complete solutions to receive full marks. 1. Short Answer Problems a) Let A = 1 0 0 1 0 0 1 1 0 0 0 . Write a basis for the Row( A ), Col( A ) and Null( A ). Solution: A basis for Row( A ) is 1 1 , 1 1 . A basis for Null( A ) is 1 , 1 1 1 A basis for Col( A ) is 1 , 1 . A basis for Null( A T ) is 1 . b) Determine if h p,q i = p ( 1) q ( 1) + p (1) q (1) is an inner product for P 2 . Solution: It is not since h x 2 1 ,x 2 1 i = 0 but x 2 1 6 = ~ 0 in P 2 . c) State the RankNullity Theorem. Solution: Suppose that V is an ndimensional vector space and that L : V → W is a linear mapping into a vector space W . Then rank( L ) + nullity( L ) = n . d) Find the rank and nullity of the linear mapping T : P 2 → M 2 × 2 ( R ) defined by T ( a + bx + cx 2 ) = c b c . Solution: rank( T ) = 2, nullity( T ) = 1....
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This note was uploaded on 09/30/2011 for the course MATH 235 taught by Professor Celmin during the Spring '08 term at Waterloo.
 Spring '08
 CELMIN
 Math

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