Let T1, T2 be linear transformations given by (;)-( 7: ***) (;))-(- Зх + 5у T1 T2 -2x + 7y ) 5y Find the matrix A such that corresponds T1(T2(x)) = Ax.
Let T1, T2 be linear transformations given by (;)-( 7: ***) (;))-(- Зх + 5у T1 T2 -2x + 7y ) 5y Find the matrix A such that corresponds T1(T2(x)) = Ax.
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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Let T1, T2 be linear transformations given by... (see image)....
Find the matrix A such that.... (see image)
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# Linear Transformations and Matrices
## Problem Statement
Let \( T_1, T_2 \) be linear transformations given by
\[
T_1 \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \left( \begin{array}{c} 3x + 5y \\ -2x + 7y \end{array} \right)
\]
\[
T_2 \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \left( \begin{array}{c} -2x + 9y \\ 5y \end{array} \right)
\]
Find the matrix \( A \) such that corresponds \( T_1(T_2(x)) = Ax \).
---
This problem involves finding a matrix that represents the composition of two linear transformations. The goal is to determine a single matrix \( A \) that performs the operation \( T_1 \) after \( T_2 \) has been applied to a vector \( x \).
### Graphical Explanation of Transformations
1. **First Transformation \( T_2 \)**:
- The vector \( \begin{bmatrix} x \\ y \end{bmatrix} \) is transformed into \( \left( \begin{array}{c} -2x + 9y \\ 5y \end{array} \right) \).
2. **Second Transformation \( T_1 \)**:
- The output from \( T_2 \), which is \( \left( \begin{array}{c} -2x + 9y \\ 5y \end{array} \right) \), is then transformed again using \( T_1 \).
### Detailed Steps
1. **Compute \( T_2(x) \)**:
For a vector \( \begin{bmatrix} x \\ y \end{bmatrix} \):
\( T_2 \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \left( \begin{array}{c} -2x + 9y \\ 5y \end{array} \right) \)
2. **Apply \( T_1 \) to the Result of \( T_2 \)**:
Let \( u](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcc17e2a4-76f7-46f2-97e3-7d367a16a370%2F94db7eac-fbfe-4590-b18e-76db2c9b6090%2F7tsjvj.png&w=3840&q=75)
Transcribed Image Text:---
# Linear Transformations and Matrices
## Problem Statement
Let \( T_1, T_2 \) be linear transformations given by
\[
T_1 \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \left( \begin{array}{c} 3x + 5y \\ -2x + 7y \end{array} \right)
\]
\[
T_2 \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \left( \begin{array}{c} -2x + 9y \\ 5y \end{array} \right)
\]
Find the matrix \( A \) such that corresponds \( T_1(T_2(x)) = Ax \).
---
This problem involves finding a matrix that represents the composition of two linear transformations. The goal is to determine a single matrix \( A \) that performs the operation \( T_1 \) after \( T_2 \) has been applied to a vector \( x \).
### Graphical Explanation of Transformations
1. **First Transformation \( T_2 \)**:
- The vector \( \begin{bmatrix} x \\ y \end{bmatrix} \) is transformed into \( \left( \begin{array}{c} -2x + 9y \\ 5y \end{array} \right) \).
2. **Second Transformation \( T_1 \)**:
- The output from \( T_2 \), which is \( \left( \begin{array}{c} -2x + 9y \\ 5y \end{array} \right) \), is then transformed again using \( T_1 \).
### Detailed Steps
1. **Compute \( T_2(x) \)**:
For a vector \( \begin{bmatrix} x \\ y \end{bmatrix} \):
\( T_2 \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \left( \begin{array}{c} -2x + 9y \\ 5y \end{array} \right) \)
2. **Apply \( T_1 \) to the Result of \( T_2 \)**:
Let \( u
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