Given the ordered basis B=A₁ -1 0 -[5], 42 - [8 A₂ -2 6 (4]} 8 16 -33 -1 3 [¹].4 A4 0 -9 A3 - find [A]B, the coordinates of A [V]B = Ex: 5 9 5 -19 37 with respect to B. 4
Given the ordered basis B=A₁ -1 0 -[5], 42 - [8 A₂ -2 6 (4]} 8 16 -33 -1 3 [¹].4 A4 0 -9 A3 - find [A]B, the coordinates of A [V]B = Ex: 5 9 5 -19 37 with respect to B. 4
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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Question
![**Solution to Coordinate Transformation Problem**
Given the ordered basis \( \mathcal{B} = \{ A_1, A_2, A_3, A_4 \} \):
- \( A_1 = \begin{bmatrix} -1 & -5 \\ 4 & -2 \end{bmatrix} \)
- \( A_2 = \begin{bmatrix} 0 & 1 \\ 9 & 5 \end{bmatrix} \)
- \( A_3 = \begin{bmatrix} -1 & 3 \\ 0 & -9 \end{bmatrix} \)
- \( A_4 = \begin{bmatrix} 6 & 4 \\ 8 & -8 \end{bmatrix} \)
Find \( [A]_{\mathcal{B}} \), the coordinates of \( A = \begin{bmatrix} -16 & -19 \\ -33 & 37 \end{bmatrix} \) with respect to \( \mathcal{B} \).
**Task Explanation:**
This problem requires finding the coordinates of matrix \( A \) in terms of the ordered basis \( \mathcal{B} \). The coordinates represent how to express matrix \( A \) as a linear combination of the given basis matrices \( A_1, A_2, A_3, A_4 \).
\[ [v]_{\mathcal{B}} = \begin{bmatrix} \text{Ex: 5} \\ \text{(empty)} \\ \text{(empty)} \\ \text{(empty)} \end{bmatrix} \]
**Instructions:**
1. Set up an equation expressing \( A \) as a linear combination of the basis matrices.
2. Solve for the coefficients that represent \( [A]_{\mathcal{B}} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97fa71a9-ddb9-496b-9b0a-bf970e388fad%2F907fb342-9943-4c16-9198-cf7befbd0715%2F1v3yph_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Solution to Coordinate Transformation Problem**
Given the ordered basis \( \mathcal{B} = \{ A_1, A_2, A_3, A_4 \} \):
- \( A_1 = \begin{bmatrix} -1 & -5 \\ 4 & -2 \end{bmatrix} \)
- \( A_2 = \begin{bmatrix} 0 & 1 \\ 9 & 5 \end{bmatrix} \)
- \( A_3 = \begin{bmatrix} -1 & 3 \\ 0 & -9 \end{bmatrix} \)
- \( A_4 = \begin{bmatrix} 6 & 4 \\ 8 & -8 \end{bmatrix} \)
Find \( [A]_{\mathcal{B}} \), the coordinates of \( A = \begin{bmatrix} -16 & -19 \\ -33 & 37 \end{bmatrix} \) with respect to \( \mathcal{B} \).
**Task Explanation:**
This problem requires finding the coordinates of matrix \( A \) in terms of the ordered basis \( \mathcal{B} \). The coordinates represent how to express matrix \( A \) as a linear combination of the given basis matrices \( A_1, A_2, A_3, A_4 \).
\[ [v]_{\mathcal{B}} = \begin{bmatrix} \text{Ex: 5} \\ \text{(empty)} \\ \text{(empty)} \\ \text{(empty)} \end{bmatrix} \]
**Instructions:**
1. Set up an equation expressing \( A \) as a linear combination of the basis matrices.
2. Solve for the coefficients that represent \( [A]_{\mathcal{B}} \).
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