Find the standard matrices A and A' for T = T₂ o T₁ and T' = T₁ 0 T₂. T₁: R² → R², T₁(x, y) = (x - 5y, 2x + 3y) T₂: R² R², T₂(x, y) = (0, x) w A = Jama A' = E SHE
Find the standard matrices A and A' for T = T₂ o T₁ and T' = T₁ 0 T₂. T₁: R² → R², T₁(x, y) = (x - 5y, 2x + 3y) T₂: R² R², T₂(x, y) = (0, x) w A = Jama A' = E SHE
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![**Finding the Standard Matrices for Composite Transformations**
In this section, we will learn how to find the standard matrices \( A \) and \( A' \) for the transformations \( T = T_2 \circ T_1 \) and \( T' = T_1 \circ T_2 \).
### Given Transformations:
1. \( T_1: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \), with \( T_1(x, y) = (x - 5y, 2x + 3y) \)
2. \( T_2: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \), with \( T_2(x, y) = (0, x) \)
### Tasks:
- Find the standard matrix \( A \) for transformation \( T = T_2 \circ T_1 \)
- Find the standard matrix \( A' \) for transformation \( T' = T_1 \circ T_2 \)
### Matrix Representations:
We need to represent the transformations as matrices. Here are the forms:
\[ A =
\begin{bmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{bmatrix}
\]
\[ A' =
\begin{bmatrix}
a'_{11} & a'_{12} \\
a'_{21} & a'_{22}
\end{bmatrix}
\]
Given the definitions of \( T_1 \) and \( T_2 \), these matrices will be constructed based on the linear combinations of \( x \) and \( y \).
### Explanation of Graphical Elements:
The image shows placeholders for the elements of matrices \( A \) and \( A' \). There are also arrows indicating the placement of the elements in the 2x2 matrices. The green arrows show where to input the components derived from the transformation formulas.
By following the provided transformation formulas \( T_1(x, y) \) and \( T_2(x, y) \), you will systematically fill in the elements of \( A \) and \( A' \):
#### For \( A \) (as \( T = T_2 \circ T_1 \)):
1. Calculate how \( T_1(x, y) \) is mapped](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F74e12d39-e69c-4fbc-93cf-0079e6a76190%2Fdeefd3ba-b1fe-43f2-9ee6-0509d025aed7%2Fsjjq4zoe_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Standard Matrices for Composite Transformations**
In this section, we will learn how to find the standard matrices \( A \) and \( A' \) for the transformations \( T = T_2 \circ T_1 \) and \( T' = T_1 \circ T_2 \).
### Given Transformations:
1. \( T_1: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \), with \( T_1(x, y) = (x - 5y, 2x + 3y) \)
2. \( T_2: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \), with \( T_2(x, y) = (0, x) \)
### Tasks:
- Find the standard matrix \( A \) for transformation \( T = T_2 \circ T_1 \)
- Find the standard matrix \( A' \) for transformation \( T' = T_1 \circ T_2 \)
### Matrix Representations:
We need to represent the transformations as matrices. Here are the forms:
\[ A =
\begin{bmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{bmatrix}
\]
\[ A' =
\begin{bmatrix}
a'_{11} & a'_{12} \\
a'_{21} & a'_{22}
\end{bmatrix}
\]
Given the definitions of \( T_1 \) and \( T_2 \), these matrices will be constructed based on the linear combinations of \( x \) and \( y \).
### Explanation of Graphical Elements:
The image shows placeholders for the elements of matrices \( A \) and \( A' \). There are also arrows indicating the placement of the elements in the 2x2 matrices. The green arrows show where to input the components derived from the transformation formulas.
By following the provided transformation formulas \( T_1(x, y) \) and \( T_2(x, y) \), you will systematically fill in the elements of \( A \) and \( A' \):
#### For \( A \) (as \( T = T_2 \circ T_1 \)):
1. Calculate how \( T_1(x, y) \) is mapped
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