Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Example 1
**Problem:**
Find the matrix \( A' \) for \( T \) relative to the basis \( B' \).
- **Transformation:** \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \), defined as \( T(x, y) = (x - y, y - 2x) \).
- **Basis \( B' \):** \(\{(1, -2), (0, 3)\}\)
**Attempted Solution:**
\[ A' = \begin{bmatrix} 3 & -3 \\ 11/3 & -4 \end{bmatrix} \]
- The result here is marked with a red cross, indicating the matrix \( A' \) is incorrect.
### Example 2
**Problem:**
Find the matrix \( A' \) for \( T \) relative to the basis \( B' \).
- **Transformation:** \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \), defined as \( T(x, y) = (-7x + y, 7x - y) \).
- **Basis \( B' \):** \(\{(1, -1), (-1, 5)\}\)
**Attempted Solution:**
\[ A' = \begin{bmatrix} 48 & -72 \\ -16 & 24 \end{bmatrix} \]
- The result here is marked with a red cross, indicating the matrix \( A' \) is incorrect.
### Notes
In both examples, the transformation and the basis for the transformation are given, but errors are identified in the attempted matrix calculations. Each solution attempt results in an incorrect matrix, as indicated by the red crosses.](https://content.bartleby.com/qna-images/question/8b8a4ad9-b66e-4dbe-8b3f-39bf0adb3cae/ca62cbf9-a882-4738-8fb8-50d94d2157a0/vqucb4d_processed.png)
Transcribed Image Text:### Example 1
**Problem:**
Find the matrix \( A' \) for \( T \) relative to the basis \( B' \).
- **Transformation:** \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \), defined as \( T(x, y) = (x - y, y - 2x) \).
- **Basis \( B' \):** \(\{(1, -2), (0, 3)\}\)
**Attempted Solution:**
\[ A' = \begin{bmatrix} 3 & -3 \\ 11/3 & -4 \end{bmatrix} \]
- The result here is marked with a red cross, indicating the matrix \( A' \) is incorrect.
### Example 2
**Problem:**
Find the matrix \( A' \) for \( T \) relative to the basis \( B' \).
- **Transformation:** \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \), defined as \( T(x, y) = (-7x + y, 7x - y) \).
- **Basis \( B' \):** \(\{(1, -1), (-1, 5)\}\)
**Attempted Solution:**
\[ A' = \begin{bmatrix} 48 & -72 \\ -16 & 24 \end{bmatrix} \]
- The result here is marked with a red cross, indicating the matrix \( A' \) is incorrect.
### Notes
In both examples, the transformation and the basis for the transformation are given, but errors are identified in the attempted matrix calculations. Each solution attempt results in an incorrect matrix, as indicated by the red crosses.
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