Suppose A is the matrix for T: R - R relative to the standard basis. Find the diagonal matrix A' for T relative to the basis B'. -1 -2 0 A = -1 0 0 0 2 B' = {(-1, 1, 0), (2, 1, 0), (0, 0, 1)} A'=
Suppose A is the matrix for T: R - R relative to the standard basis. Find the diagonal matrix A' for T relative to the basis B'. -1 -2 0 A = -1 0 0 0 2 B' = {(-1, 1, 0), (2, 1, 0), (0, 0, 1)} A'=
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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Suppose A is the matrix for T: R3 -> R3 relative to the standar basis. Find the diagonal matrix A' for T relative ot the basis B'.
![**Problem Statement**
Suppose \( A \) is the matrix for \( T: \mathbb{R}^3 \to \mathbb{R}^3 \) relative to the standard basis. Find the diagonal matrix \( A' \) for \( T \) relative to the basis \( B' \).
\[
A = \begin{bmatrix}
-1 & -2 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 2
\end{bmatrix}
\]
\[
B' = \{ (-1, 1, 0), (2, 1, 0), (0, 0, 1) \}
\]
**Solution**
A step-by-step approach is needed to find the diagonal matrix \( A' \).
**Explanation of Matrix Layout**
- \( A \) is a \( 3 \times 3 \) matrix given as part of the problem.
- The basis \( B' \) is composed of vectors \((-1, 1, 0)\), \( (2, 1, 0) \), and \( (0, 0, 1) \).
**Tasks**
- Calculate \( A' \) by finding the change of basis from the standard basis to \( B' \) and expressing \( A \) in this new basis as a diagonal matrix.
**Matrix Diagram**
The matrix \( A' \) is represented as a blank \( 3 \times 3 \) grid, indicating where the elements of the diagonal matrix will be placed. The arrows suggest a method or process for filling in these elements, possibly signifying transformation or eigenvalue computation steps.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7d6fc442-7ad3-4518-9add-f310e8d32dde%2F040a4a61-a1e8-4b3d-8723-bf042b8c9681%2Fun8l1h9_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Suppose \( A \) is the matrix for \( T: \mathbb{R}^3 \to \mathbb{R}^3 \) relative to the standard basis. Find the diagonal matrix \( A' \) for \( T \) relative to the basis \( B' \).
\[
A = \begin{bmatrix}
-1 & -2 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 2
\end{bmatrix}
\]
\[
B' = \{ (-1, 1, 0), (2, 1, 0), (0, 0, 1) \}
\]
**Solution**
A step-by-step approach is needed to find the diagonal matrix \( A' \).
**Explanation of Matrix Layout**
- \( A \) is a \( 3 \times 3 \) matrix given as part of the problem.
- The basis \( B' \) is composed of vectors \((-1, 1, 0)\), \( (2, 1, 0) \), and \( (0, 0, 1) \).
**Tasks**
- Calculate \( A' \) by finding the change of basis from the standard basis to \( B' \) and expressing \( A \) in this new basis as a diagonal matrix.
**Matrix Diagram**
The matrix \( A' \) is represented as a blank \( 3 \times 3 \) grid, indicating where the elements of the diagonal matrix will be placed. The arrows suggest a method or process for filling in these elements, possibly signifying transformation or eigenvalue computation steps.
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