Explanation Given information : Consider the basis of U 2 × 2 is ℵ = [ ( 1 0 0 0 ) , ( 0 1 0 0 ) , ( 0 0 0 1 ) ] . Consider the basis β = [ ( 1 0 0 1 ) , ( 0 1 0 0 ) , ( 1 0 0 − 1 ) ] in exercise 7 . Theorem 4.3.4 Change of the basis in a subspace of ℝ n . Consider a subspace V of ℝ n with two bases ℵ = ( a → 1 , ... , a → n ) and β = ( b → 1 , ... , b → n ) .Let S be the change of basis matrix from β to ℵ .Then the following equation holds: [ . . . . . . . . b → 1 . . b → n . . . . ] = [ . . . . . . . . a → 1 . . a → n . . . . ] S Calculation: Consider the transformation T ( M ) = M [ 1 2 0 1 ] − [ 1 2 0 1 ] M in exercise 7 . For the given basis ℵ = [ ( 1 0 0 0 ) , ( 0 1 0 0 ) , ( 0 0 0 1 ) ] , the matrix A can be found as shown. T ( 1 0 0 0 ) = ( 1 0 0 0 ) [ 1 2 0 1 ] − [ 1 2 0 1 ] ( 1 0 0 0 ) = [ 1 2 0 0 ] − [ 1 0 0 0 ] = [ 0 2 0 0 ] = 2. [ 0 1 0 0 ] T ( 0 1 0 0 ) = ( 0 1 0 0 ) [ 1 2 0 1 ] − [ 1 2 0 1 ] ( 0 1 0 0 ) = [ 0 1 0 0 ] − [ 0 1 0 0 ] = [ 0 0 0 0 ] T ( 0 0 0 1 ) = ( 0 0 0 1 ) [ 1 2 0 1 ] − [ 1 2 0 1 ] ( 0 0 0 1 ) = [ 0 0 0 1 ] − [ 0 2 0 1 ] = [ 0 − 2 0 0 ] = − 2 [ 0 1 0 0 ] Therefore, A = ( 0 0 0 2 0 − 2 0 0 0 ) Consider the transformation T ( M ) = M [ 1 2 0 1 ] − [ 1 2 0 1 ] M

Linear Algebra with Applications (2-Download)

5th Edition

ISBN: 9780321796974

Author: Otto Bretscher

Publisher: PEARSON

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Chapter 4.3, Problem 8E

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To Find: the matrices A and B with respect to the given two bases

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Chapter 4.3, Problem 7E

Chapter 4.3, Problem 9E

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Chapter 4 Solutions

Linear Algebra with Applications (2-Download)

Ch. 4.1 - GOAL Find a basis of a linear space and thus...Ch. 4.1 - GOAL Find a basis of a linear space and thus...Ch. 4.1 - GOAL Find a basis of a linear space and thus...Ch. 4.1 - GOAL Find a basis of a linear space and thus...Ch. 4.1 - GOAL Find a basis of a linear space and thus...Ch. 4.1 - Which of the subsets V of 33given in Exercises 6...Ch. 4.1 - Which of the subsets V of 33given in Exercises 6...Ch. 4.1 - Which of the subsets V of 33given in Exercises 6...Ch. 4.1 - Which of the subsets V of 33given in Exercises 6...Ch. 4.1 - Which of the subsets V of 33given in Exercises 6...

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the matrices A and B with respect to the given two bases

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To Find: the matrices A and B with respect to the given two bases

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