a) T: R² → R², T (C)=₁ T(B) = B T([*]) = 3 b) Find a 2x2 matrix 2] [21] [8813-3 = 25 Previous
a) T: R² → R², T (C)=₁ T(B) = B T([*]) = 3 b) Find a 2x2 matrix 2] [21] [8813-3 = 25 Previous
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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![Sure, here's the transcription:
---
### Linear Transformation and Matrix Representation
#### a) Transformation Definition
Consider the linear transformation \( T: \mathbb{R}^2 \to \mathbb{R}^2 \), defined as follows:
\[
T \left( \begin{bmatrix} 1 \\ 2 \end{bmatrix} \right) = \begin{bmatrix} 2 \\ 1 \end{bmatrix}
\]
\[
T \left( \begin{bmatrix} 2 \\ 5 \end{bmatrix} \right) = \begin{bmatrix} 1 \\ 3 \end{bmatrix}
\]
To find \( T \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) \), determine the transformation matrix.
\[
T \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} \square \\ \square \end{bmatrix}
\]
#### b) Find a 2x2 Matrix
Determine the 2x2 matrix that represents this linear transformation such that:
\[
\begin{bmatrix} \square & \square \\ \square & \square \end{bmatrix}
\begin{bmatrix} 1 & 2 \\ 2 & 5 \end{bmatrix} = \begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix}
\]
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9978ffa8-8e6a-4550-8363-e044b6a6e895%2Fcade010c-c36c-433a-bbd5-a4baa2866b94%2Fzema5wq_processed.png&w=3840&q=75)
Transcribed Image Text:Sure, here's the transcription:
---
### Linear Transformation and Matrix Representation
#### a) Transformation Definition
Consider the linear transformation \( T: \mathbb{R}^2 \to \mathbb{R}^2 \), defined as follows:
\[
T \left( \begin{bmatrix} 1 \\ 2 \end{bmatrix} \right) = \begin{bmatrix} 2 \\ 1 \end{bmatrix}
\]
\[
T \left( \begin{bmatrix} 2 \\ 5 \end{bmatrix} \right) = \begin{bmatrix} 1 \\ 3 \end{bmatrix}
\]
To find \( T \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) \), determine the transformation matrix.
\[
T \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} \square \\ \square \end{bmatrix}
\]
#### b) Find a 2x2 Matrix
Determine the 2x2 matrix that represents this linear transformation such that:
\[
\begin{bmatrix} \square & \square \\ \square & \square \end{bmatrix}
\begin{bmatrix} 1 & 2 \\ 2 & 5 \end{bmatrix} = \begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix}
\]
---
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