Explanation Given: The given condition is: T : P 1 → P 1 : T ( a + b x ) = a + ( a + 2 b ) x Procedure used: A linear transformation; T : V → V There exist a basis B for V such that the matrix for T relative to B is diagonal when the standard matrix for T is diagonalizable. Finding eigenvalues and eigenvectors: Let A be an n × n matrix. 1. Form the characteristics equation | λ I − A | = 0 . 2. Find the real roots of the characteristics equation. These are the eigenvalues of A . 3. For each eigenvalue λ i , find the eigenvectors corresponding to λ i by solving the homogeneous system ( λ i I − A ) X = 0 . This can be required row reducing an n × n matrix. Sufficient condition for diagonalization: If an n × n matrix A has n distinct eigenvalues, then the corresponding eigenvalues are linearly independent and A is diagonalizable. Calculation: Consider the given condition: T : P 1 → P 1 : T ( a + b x ) = a + ( a + 2 b ) x Consider { 1 , x } as a standard basis of P 1 . Substitute in the above condition, we get: T ( 1 ) = 1 + x = 1 ⋅ 1 + 1 ⋅ x and, T ( x ) = 2 x = 0 ⋅ 1 + 2 ⋅ x The standard matrix for T is: A = [ 1 0 1 2 ] The roots of the characteristic equation are: | [ 1 0 1 2 ] − λ [ 1 0 0 1 ] | = 0 | [ 1 0 1 2 ] − [ λ 0 0 λ ] | = 0 | [ 1 − λ 0 1 2 − λ ] | = 0 ( 1 − λ ) ( 2 − λ ) = 0 Further simplify as: ( 1 − λ ) ( 2 − λ ) = 0 λ = 1 , 2 Find the eigenvector corresponding to eigenvalue λ = 1 by solving the homogeneous system ( A − I ) X = 0 : Let X = [ x 1 x 2 ]

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ISBN: 9780357156100

Author: Ron Larson

Publisher: Cengage Limited

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Chapter 7.2, Problem 29E

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The basis B for the domain of T such that the matrix for T relative to B is diagonal for the given condition T:P1→P1:T(a+bx)=a+(a+2b)x.

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Chapter 7.2, Problem 28E

Chapter 7.2, Problem 30E

Chapter 7 Solutions

Elementary Linear Algebra

Ch. 7.1 - Verifying Eigenvalues and Eigenvectors in...Ch. 7.1 - Verifying Eigenvalues and EigenvectorsIn Exercises...Ch. 7.1 - Verifying Eigenvalues and EigenvectorsIn Exercises...Ch. 7.1 - Verifying Eigenvalues and Eigenvectors in...Ch. 7.1 - Verifying Eigenvalues and EigenvectorsIn Exercises...Ch. 7.1 - Verifying Eigenvalues and EigenvectorsIn Exercises...Ch. 7.1 - Prob. 7E Ch. 7.1 - Prob. 8E Ch. 7.1 - Determining Eigenvectors In Exercise 9-12,...Ch. 7.1 - Determining Eigenvectors In Exercise 9-12,...

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The basis B for the domain of T such that the matrix for T relative to B is diagonal for the given condition T : P 1 → P 1 : T ( a + b x ) = a + ( a + 2 b ) x .

Solution Summary: The author explains the basis B for the domain of T such that the matrix for T relative to B is diagonal.

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To find:

The basis B for the domain of T such that the matrix for T relative to B is diagonal for the given condition T:P1→P1:T(a+bx)=a+(a+2b)x.

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Chapter 7 Solutions

The basis B for the domain of T such that the matrix for T relative to B is diagonal for the given condition T : P 1 → P 1 : T ( a + b x ) = a + ( a + 2 b ) x .

Solution Summary: The author explains the basis B for the domain of T such that the matrix for T relative to B is diagonal.

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To find: The basis B for the domain of T such that the matrix for T relative to B is diagonal for the given condition T:P1→P1:T(a+bx)=a+(a+2b)x.

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Chapter 7 Solutions

To find:

The basis B for the domain of T such that the matrix for T relative to B is diagonal for the given condition T:P1→P1:T(a+bx)=a+(a+2b)x.