Math 54 - Quiz 05 Ans, Spring 2006

Math 54 - Quiz 05 Ans, Spring 2006 - P 3 . Write your...

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Math 54, Quiz 5 Solutions Sections 202/203 GSI Carter 1. For each of the following statements, write the word “true” or “false.” (a) Let T 1 : U V and T 2 : V W be two linear transformations. Then the function S from U to W defined by S ( u ) = T 2 ( T 1 ( u )) for any u U is a linear transformation. True, since T 2 ( T 1 ( u + v )) = T 2 ( T 1 ( u ) + T 1 ( v )) = T 2 ( T 1 ( u )) + T 2 ( T 1 ( v )) and T 2 ( T 1 ( α u )) = T 2 ( αT 1 ( u )) = αT 2 ( T 1 ( u )). (b) The function E : C [ a, b ] R given by E ( f ) = R b a x · f ( x ) dx for any f C [ a, b ] is a linear transformation. True, since E ( f + g ) = Z b a x [ f ( x ) + g ( x )] dx = Z b a [ x · f ( x ) + x · g ( x )] dx = Z b a x · f ( x ) dx + Z b a x · g ( x ) dx = E ( f ) + E ( g ) , and E ( αf ) = Z b a x [ αf ( x )] dx = α Z b a x · f ( x ) dx = αE ( f ) . 2. Give the coordinates of the polynomial p ( x ) = 1 - x + x 3 in terms of the basis { 1 , x - 1 , ( x - 1) 2 , ( x - 1) 3 } of the vector space
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Unformatted text preview: P 3 . Write your answer in the form of a column vector. We have to find scalars a , b , c , and d such that a + b ( x-1) + c ( x 2-2 x + 1) + d ( x 3-3 x 2 + 3 x-1) = 1-x + x 3 . This can be done by solving the system 1-1 1-1 1 1-2 3-1 1-3 1 1 which row reduces to 1 1 1 2 1 3 1 1 . Therefore, the coordinate vector is: 1 2 3 1 . 1...
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