Explanation Given: The vector is, V = P 1 ( ℝ ) ; B = { 7 − 4 x , 5 x } ; C = { 1 − 2 x , 2 + x } ; p ( x ) = 6 − 4 x . Approach: Let V be an n -dimensional vector space with ordered bases B and C . We 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, we determine the components of each vector in the “old basis” B with respect to the “new basis” C and write the component vectors in the columns of the change-of-basis matrix. Calculation: Consider the component of ( 6 − 4 x ) relative to the ordered basis B . 6 − 4 x = c 1 ( 7 − 4 x ) + c 2 ( 5 x ) = ( 7 c 1 ) + ( − 4 c 1 + 5 c 2 ) x Compare both the sides, 6 = 7 c 1 ... ( 1 ) − 4 = − 4 c 1 + 5 c 2 ... ( 2 ) Solve equation ( 1 ) and equation ( 2 ) . Then, The value c 1 = 6 7 and c 2 = − 4 35 . Thus, the component is [ v ] B = [ 6 7 − 4 35 ] . Consider the component of ( 6 − 4 x ) relative to the ordered basis C . 6 − 4 x = c 1 ( 1 − 2 x ) + c 2 ( 2 + x ) = ( c 1 + 2 c 2 ) + ( − 2 c 1 + c 2 ) x Compare both the sides, 6 = c 1 + 2 c 2

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 35P

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To verify:

The equation [v]C=PC←B[v]B.

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Chapter 4.7, Problem 34P

Chapter 4.7, Problem 36P

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To verify: The equation [ v ] C = P C ← B [ v ] B .

Solution Summary: The author explains how to verify the equation left[vright]_C=P

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To verify:

The equation [v]C=PC←B[v]B.

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Chapter 4 Solutions

To verify: The equation [ v ] C = P C ← B [ v ] B .

Solution Summary: The author explains how to verify the equation left[vright]_C=P

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To verify: The equation [v]C=PC←B[v]B.

Want to see the full answer?

Chapter 4 Solutions

To verify:

The equation [v]C=PC←B[v]B.