# Linear Transformation Problem: Finding Standard MatricesIn this exercise, we aim to determine the standard matrices \( A \) and \( A' \) for the composed transformations \( T = T_2 \circ T_1 \) and \( T' = T_1 \circ T_2 \).### Given Transformations- **Transformation \( T_1 \):** \( T_1: \mathbb{R}^2 \to \mathbb{R}^2 \) Defined by: \[ T_1(x, y) = (x - 3y, 3x + 3y) \]- **Transformation \( T_2 \):** \( T_2: \mathbb{R}^2 \to \mathbb{R}^2 \) Defined by: \[ T_2(x, y) = (0, x) \]### Task- **Find the standard matrix \( A \) for \( T = T_2 \circ T_1 \).** The composition implies first applying \( T_1 \), then \( T_2 \).- **Find the standard matrix \( A' \) for \( T' = T_1 \circ T_2 \).** The composition implies first applying \( T_2 \), then \( T_1 \).### Diagram Explanation- The diagram displays two blank 2x2 matrices labeled \( A \) and \( A' \). These matrices need to be filled with the coefficients representing the standard matrices of the transformations \( T \) and \( T' \).- **\( A \)** and **\( A' \)** each consist of: - Two rows of two placeholders. - Green arrows pointing right signify the positions to fill for the output vector. - Green arrows pointing down indicate input dimensions processed during matrix multiplication.The exercise requires calculating and inputting the correct values in these matrices based on the composition of \( T_1 \) and \( T_2 \).

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Answered: Find the standard matrices A and A' for…

Find the standard matrices A and A' for T = T2 ° T1 and T' = T1 ° T2. T1: R2 → R2, T;(x, y) = (x – 3y, 3x + 3y) T2: R2 → R², T2(x, y) = (0, x) A = A' =

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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Question

# Linear Transformation Problem: Finding Standard Matrices

In this exercise, we aim to determine the standard matrices \( A \) and \( A' \) for the composed transformations \( T = T_2 \circ T_1 \) and \( T' = T_1 \circ T_2 \).

### Given Transformations

- **Transformation \( T_1 \):**

\( T_1: \mathbb{R}^2 \to \mathbb{R}^2 \)

Defined by:
\[
T_1(x, y) = (x - 3y, 3x + 3y)
\]

- **Transformation \( T_2 \):**

\( T_2: \mathbb{R}^2 \to \mathbb{R}^2 \)

Defined by:
\[
T_2(x, y) = (0, x)
\]

### Task

- **Find the standard matrix \( A \) for \( T = T_2 \circ T_1 \).**

The composition implies first applying \( T_1 \), then \( T_2 \).

- **Find the standard matrix \( A' \) for \( T' = T_1 \circ T_2 \).**

The composition implies first applying \( T_2 \), then \( T_1 \).

### Diagram Explanation

- The diagram displays two blank 2x2 matrices labeled \( A \) and \( A' \). These matrices need to be filled with the coefficients representing the standard matrices of the transformations \( T \) and \( T' \).

- **\( A \)** and **\( A' \)** each consist of:
- Two rows of two placeholders.
- Green arrows pointing right signify the positions to fill for the output vector.
- Green arrows pointing down indicate input dimensions processed during matrix multiplication.

The exercise requires calculating and inputting the correct values in these matrices based on the composition of \( T_1 \) and \( T_2 \).

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