Explanation Given: The vector is, V = M 2 ( ℝ ) ; B = { E 12 , E 22 , E 21 , E 11 } ; C = { E 22 , E 11 , E 21 , E 12 } . Approach: Let V be an n -dimensional vector space with ordered bases B and C . Define the change-of-basis matrix from B to C by P C ← B = [ [ v 1 ] c , [ v 2 ] c , … , [ v n ] c ] In words, determine the components of each vector in the “old basis” B with respect the “new basis” C and write the component vectors in the columns of the change-of-basis matrix. Calculation: Consider the component of B E 22 or [ 0 0 0 1 ] relative to the ordered basis C . [ 0 0 0 1 ] = c 1 [ 0 1 0 0 ] + c 2 [ 0 0 0 1 ] + c 3 [ 0 0 1 0 ] + c 4 [ 1 0 0 0 ] The system of equation can be written as, 1 = c 2 0 = c 4 0 = c 3 0 = c 1 Thus, the component is [ E 22 ] C = [ 0 1 0 0 ] . Consider the component of B E 11 or [ 1 0 0 0 ] relative to the ordered basis C . [ 1 0 0 0 ] = c 1 [ 0 1 0 0 ] + c 2 [ 0 0 0 1 ] + c 3 [ 0 0 1 0 ] + c 4 [ 1 0 0 0 ] The system of equation can be written as, 0 = c 2 1 = c 4 0 = c 3 0 = c 1 Thus, the component is [ E 11 ] C = [ 0 0 0 1 ]

Differential Equations and Linear Algebra (4th Edition)

4th Edition

ISBN: 9780321964670

Author: Stephen W. Goode, Scott A. Annin

Publisher: PEARSON

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Chapter 4.7, Problem 32P

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The change-of-basis matrix PC←B.

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Differential Equations and Linear Algebra (4th Edition)

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The change-of-basis matrix P C ← B .

Solution Summary: The author explains how to determine the change-of-basis matrix from V to C by determining the components of each vector in the "old basis" B.

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The change-of-basis matrix PC←B.

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