### 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.### NotesIn 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.

Answered: Find the matrix A' for T relative to…

Find the matrix A' for T relative to the basis B'. A' = T: R² → R², T(x, y) = (x - y, y - 2x), B' = {(1, -2), (0, 3)} 3 11/3 -3 -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

### 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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Given:

$T : ℝ^{2} \to ℝ^{2}$ defined by $T (x, y) = (x - y, y - 2 x)$ .
Basis $B' = \{(1, - 2), (0, 3)\}$ .

To find:

Matrix $A'$ for $T$ relative to the basis $B'$ .

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