Let P denote a triangle with the following vertices: A = [6,1] B = [7,4] C = [10, 2] Give an example of a matrix M such that the linear transformation T: R² → R² given by T (v) = Mv satisfies the following conditions: • T 2 ([J]) = [3] 6 • T transforms the triangle P into a triangle of area 8. 6 4 2 0 4 A 6 B 8 10 с 12
Let P denote a triangle with the following vertices: A = [6,1] B = [7,4] C = [10, 2] Give an example of a matrix M such that the linear transformation T: R² → R² given by T (v) = Mv satisfies the following conditions: • T 2 ([J]) = [3] 6 • T transforms the triangle P into a triangle of area 8. 6 4 2 0 4 A 6 B 8 10 с 12
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
![**Triangle with Linear Transformation**
Let \( P \) denote a triangle with the following vertices:
\[
A = [6, 1]
\]
\[
B = [7, 4]
\]
\[
C = [10, 2]
\]
Give an example of a matrix \( M \) such that the linear transformation \( T : \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) given by \( T(\mathbf{v}) = M \mathbf{v} \) satisfies the following conditions:
- \( T \left( \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right) = \begin{bmatrix} 2 \\ 6 \end{bmatrix} \)
- \( T \) transforms the triangle \( P \) into a triangle of area 8.
**Graph Description:**
The graph to the right shows triangle \( P \) on a coordinate plane. Points \( A \), \( B \), and \( C \) correspond to the vertices specified above. The triangle is shaded in orange and has edges connected by bold red lines. Each vertex is labeled accordingly, showing the location of points \( A(6,1) \), \( B(7,4) \), and \( C(10,2) \) on the graph. The background grid aids in visualizing the position and shape of the triangle within the boundary of the axes ranging from 4 to 12 on the x-axis and 0 to 6 on the y-axis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb493ecdd-cbdc-400d-a05a-de2010eb2d52%2F076e1fd5-2784-4936-ab70-4d4acd5a10f7%2Fluzmg9_processed.png&w=3840&q=75)
Transcribed Image Text:**Triangle with Linear Transformation**
Let \( P \) denote a triangle with the following vertices:
\[
A = [6, 1]
\]
\[
B = [7, 4]
\]
\[
C = [10, 2]
\]
Give an example of a matrix \( M \) such that the linear transformation \( T : \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) given by \( T(\mathbf{v}) = M \mathbf{v} \) satisfies the following conditions:
- \( T \left( \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right) = \begin{bmatrix} 2 \\ 6 \end{bmatrix} \)
- \( T \) transforms the triangle \( P \) into a triangle of area 8.
**Graph Description:**
The graph to the right shows triangle \( P \) on a coordinate plane. Points \( A \), \( B \), and \( C \) correspond to the vertices specified above. The triangle is shaded in orange and has edges connected by bold red lines. Each vertex is labeled accordingly, showing the location of points \( A(6,1) \), \( B(7,4) \), and \( C(10,2) \) on the graph. The background grid aids in visualizing the position and shape of the triangle within the boundary of the axes ranging from 4 to 12 on the x-axis and 0 to 6 on the y-axis.
Expert Solution
Step 1
M is the matrix representation of T with respect to standard order basis.
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