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Find a basis B for the domain of T such that the matrix for T relative to B is diagonal. T: R² R²: T(x, y) = (2x + y, 2x + y) B = ↓↑

Find a basis B for the domain of T such that the matrix for T relative to B is diagonal. T: R² R²: T(x, y) = (2x + y, 2x + y) B = ↓↑

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Topic: Linear Algebra - Diagonalization** **Objective:** Find a basis \( B \) for the domain of \( T \) such that the matrix for \( T \) relative to \( B \) is diagonal. **Function Definition:** \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) \( T(x, y) = (2x + y, 2x + y) \) **Problem Setup:** To solve this problem, we need to find a basis \( B \) such that when the linear transformation \( T \) is expressed relative to this basis, the resulting matrix is diagonal. **Matrix Representation:** The blank matrix in the image indicates where the basis vectors should be placed. The diagram implies the transformation and replacement operations to achieve a diagonal matrix. - The placeholders (empty rectangles) are likely representing the basis vectors that need to be determined. - Arrows indicate the transformation of the vectors under \( T \). **Approach:** 1. **Identify Eigenvectors and Eigenvalues:** - To diagonalize \( T \), first compute the eigenvalues and eigenvectors. 2. **Diagonalization Process:** - Once the eigenvectors are determined, they will form the basis \( B \). - The diagonal matrix will have eigenvalues on its main diagonal. 3. **Verification:** - Ensure the resulting matrix is diagonal when \( T \) is expressed with respect to the found basis \( B \). By completing these steps, the transformation \( T \) will be represented by a diagonal matrix, simplifying the understanding and computation of \( T \) in the context of the chosen basis.
Transcribed Image Text:**Topic: Linear Algebra - Diagonalization** **Objective:** Find a basis \( B \) for the domain of \( T \) such that the matrix for \( T \) relative to \( B \) is diagonal. **Function Definition:** \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) \( T(x, y) = (2x + y, 2x + y) \) **Problem Setup:** To solve this problem, we need to find a basis \( B \) such that when the linear transformation \( T \) is expressed relative to this basis, the resulting matrix is diagonal. **Matrix Representation:** The blank matrix in the image indicates where the basis vectors should be placed. The diagram implies the transformation and replacement operations to achieve a diagonal matrix. - The placeholders (empty rectangles) are likely representing the basis vectors that need to be determined. - Arrows indicate the transformation of the vectors under \( T \). **Approach:** 1. **Identify Eigenvectors and Eigenvalues:** - To diagonalize \( T \), first compute the eigenvalues and eigenvectors. 2. **Diagonalization Process:** - Once the eigenvectors are determined, they will form the basis \( B \). - The diagonal matrix will have eigenvalues on its main diagonal. 3. **Verification:** - Ensure the resulting matrix is diagonal when \( T \) is expressed with respect to the found basis \( B \). By completing these steps, the transformation \( T \) will be represented by a diagonal matrix, simplifying the understanding and computation of \( T \) in the context of the chosen basis.
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