Find the standard T₁: R² T₂: R² A = A' matrices A and A' for T = T₂° T₁ and T' = T₁ ° T₂. R², T₁(x, y) = (x - 4y, 4x + 2y) R², T₂(x, y) = (0, x) ↓ 1
Find the standard T₁: R² T₂: R² A = A' matrices A and A' for T = T₂° T₁ and T' = T₁ ° T₂. R², T₁(x, y) = (x - 4y, 4x + 2y) R², T₂(x, y) = (0, x) ↓ 1
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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![**Instructions to Find Standard Matrices for Transformations**
This exercise involves finding the standard matrices \( A \) and \( A' \) for the given composite linear transformations \( T = T_2 \circ T_1 \) and \( T' = T_1 \circ T_2 \).
### Given Transformations
1. **Transformation \( T_1: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \):**
- \( T_1(x, y) = (x - 4y, 4x + 2y) \)
2. **Transformation \( T_2: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \):**
- \( T_2(x, y) = (0, x) \)
### Required
**Matrix Representations:**
- **Matrix \( A \) for \( T = T_2 \circ T_1 \):**
\[
A = \begin{bmatrix}
\placeholder & \placeholder \\
\placeholder & \placeholder
\end{bmatrix}
\]
- **Matrix \( A' \) for \( T' = T_1 \circ T_2 \):**
\[
A' = \begin{bmatrix}
\placeholder & \placeholder \\
\placeholder & \placeholder
\end{bmatrix}
\]
#### Explanation:
- Each transformation is mapped from \(\mathbb{R}^2\) to \(\mathbb{R}^2\), which means they operate on two-dimensional vectors.
- Matrices \( A \) and \( A' \) should capture the effect of the transformations on these vectors.
- The placeholders in the matrices are to be filled based on the composition of the transformations \( T_1 \) and \( T_2 \).
Your task is to determine how these transformations interact when applied in the specified order, and thereby determine the resulting matrix form for each composition.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9348d578-66ea-4064-b97e-301aeae52239%2F5053d79f-3686-4c0b-bdd0-fdba8ca6d817%2Fyia9r3m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Instructions to Find Standard Matrices for Transformations**
This exercise involves finding the standard matrices \( A \) and \( A' \) for the given composite linear transformations \( T = T_2 \circ T_1 \) and \( T' = T_1 \circ T_2 \).
### Given Transformations
1. **Transformation \( T_1: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \):**
- \( T_1(x, y) = (x - 4y, 4x + 2y) \)
2. **Transformation \( T_2: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \):**
- \( T_2(x, y) = (0, x) \)
### Required
**Matrix Representations:**
- **Matrix \( A \) for \( T = T_2 \circ T_1 \):**
\[
A = \begin{bmatrix}
\placeholder & \placeholder \\
\placeholder & \placeholder
\end{bmatrix}
\]
- **Matrix \( A' \) for \( T' = T_1 \circ T_2 \):**
\[
A' = \begin{bmatrix}
\placeholder & \placeholder \\
\placeholder & \placeholder
\end{bmatrix}
\]
#### Explanation:
- Each transformation is mapped from \(\mathbb{R}^2\) to \(\mathbb{R}^2\), which means they operate on two-dimensional vectors.
- Matrices \( A \) and \( A' \) should capture the effect of the transformations on these vectors.
- The placeholders in the matrices are to be filled based on the composition of the transformations \( T_1 \) and \( T_2 \).
Your task is to determine how these transformations interact when applied in the specified order, and thereby determine the resulting matrix form for each composition.
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