Explanation Given: The linear transformation T ( x , y ) = ( y , 2 x , x + y ) . Bases B = { ( 1 , 1 ) , ( 1 , 0 ) } for R 2 and B ′ = { ( 1 , 0 , 0 ) , ( 1 , 1 , 0 ) , ( 1 , 1 , 1 ) } for R 3 . The vector ( 0 , 1 ) . Approach: Let B = { u , v } and B ′ = { u ′ , v ′ } be two bases for R 2 . If [ u ] B = [ a b ] and [ v ] B = [ c d ] , then P = [ a c b d ] is the change of coordinates matrix from B ′ to B and P − 1 is the change of coordinates matrix from B to B ′ . That is, for any v ∈ V , [ v ] B = P [ v ] B ′ [ v ] B ′ = P − 1 [ v ] B Calculation: Given linear transformation T ( x , y ) = ( y , 2 x , x + y ) . Consider the basis B = { ( 1 , 1 ) , ( 1 , 0 ) } . Then, T ( 1 , 1 ) = ( 1 , 2 × 1 , 1 + 1 ) = ( 1 , 2 , 2 ) and, T ( 1 , 0 ) = ( 0 , 2 × 1 , 1 + 0 ) = ( 0 , 2 , 1 ) Consider the basis B ′ = { ( 1 , 0 , 0 ) , ( 1 , 1 , 0 ) , ( 1 , 1 , 1 ) } . Write T ( 1 , 1 ) in terms of basis B ′ as follows: ( 1 , 2 , 2 ) = a ( 1 , 0 , 0 ) + b ( 1 , 1 , 0 ) + c ( 1 , 1 , 1 ) = ( a , 0 , 0 ) + ( b , b , 0 ) + ( c , c , c ) = ( a + b + c , b + c , c ) So,

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ISBN: 9780357156100

Author: Ron Larson

Publisher: Cengage Limited

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Chapter 7.CM, Problem 17CM

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The matrix of given linear transformation relative to given bases and use the matrix to find the image of vector (0,1).

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Chapter 7.CM, Problem 16CM

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Elementary Linear Algebra

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