2107test2sol_09

# 2107test2sol_09 - MATH 2107A TEST 2 6 1 Find a linear...

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MATH 2107A TEST 2 OCTOBER 23, 2009 1 of 2 [ 6] 1. Find a linear transformation 2 : P T with the following properties; 2 ) 2 ( = + x T , 1 ) 1 ( = T , 1 ) ( 2 = + x x T . Compute ) 2 3 ( 2 + x x T The set } , 1 , 2 { 2 x x x + + is a basis of 2 P , so every vector 2 cx bx a p + + = in 2 P is a linear combination of these vectors. i.e. ) ( ) 1 ( ) 2 ( 2 1 1 1 2 x x c b x a cx bx a p + + + + = + + = for some scalar 1 a , 1 b , 1 c . i.e. 2 1 1 1 1 1 2 ) ( ) 2 ( x c x c a b a cx bx a + + + + = + + i.e. we need to solve + c c b a c b c a b c b a 2 2 1 0 0 0 1 0 0 0 1 ~ 1 0 0 0 1 2 1 0 1 ~ 1 0 0 1 0 1 0 1 2 i.e. ) ( ) 1 )( 2 2 ( ) 2 )( ( 2 2 x x c c b a x c b cx bx a p + + + + + = + + = So, ) ( ) 1 ( ) 2 2 ( ) 2 ( ) ( ) ( 2 x x cT T c b a x T c b p T + + + + + = c a c c b a c b cx bx a T + = + + + = + + 1 . 1 ). 2 2 ( 2 ). ( ) ( 2 So, 5 3 2 ) 2 3 ( 2 = + = + x x T [9] 2. Let 4 3 : T ;) 4 3 , 2 , , 2 ( ) , , ( z y x z x z y x z y x z y x T + + + + = be a linear transformation. (a) Find ) ker( T and a basis 1 B of ) ker( T .

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2107test2sol_09 - MATH 2107A TEST 2 6 1 Find a linear...

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