5. Let A be the linear transformation R2 R2 that first reflects vectors across the line y=z, then rotates vectors 90 degrees counterclockwise abour the origin. Find the matrix representation of A.
5. Let A be the linear transformation R2 R2 that first reflects vectors across the line y=z, then rotates vectors 90 degrees counterclockwise abour the origin. Find the matrix representation of A.
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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### Linear Transformation and Matrix Representation
**Problem Statement:**
Let \( A \) be the linear transformation \( \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) that first reflects vectors across the line \( y = -x \), then rotates vectors 90 degrees counterclockwise about the origin. Find the matrix representation of \( A \).
**Solution:**
To find the matrix representation of \( A \), we need to consider the two transformations: reflection across the line \( y = -x \) and rotation by 90 degrees counterclockwise.
**Reflection across the line \( y = -x \):**
The reflection of a point \( (x, y) \) across the line \( y = -x \) can be represented by the matrix:
\[ R = \begin{pmatrix}
0 & -1 \\
-1 & 0
\end{pmatrix} \]
**Rotation by 90 degrees counterclockwise:**
The rotation of a point \( (x, y) \) by 90 degrees counterclockwise can be represented by the matrix:
\[ Q = \begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix} \]
**Combined Transformation:**
Since the transformation \( A \) first reflects vectors across the line \( y = -x \), and then rotates vectors 90 degrees counterclockwise, the combined transformation can be represented by the product of the rotation matrix \( Q \) and the reflection matrix \( R \):
\[ A = Q \cdot R \]
Calculating the product:
\[ A = \begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix}
\begin{pmatrix}
0 & -1 \\
-1 & 0
\end{pmatrix}
= \begin{pmatrix}
(-1 \cdot -1) + (0 \cdot 0) & (0 \cdot -1) + (-1 \cdot 0) \\
(1 \cdot -1) + (0 \cdot 0) & (0 \cdot -1) + (1 \cdot 0)
\end{pmatrix}
= \begin{pmatrix}
1 & 0 \\
-1 & 0
\end{pmatrix} \]
The matrix](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2e6b1be0-596d-4af1-a341-a1c152f1c814%2Febd62830-24d1-49bd-8010-47ecf6a4ac94%2F43zem7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
### Linear Transformation and Matrix Representation
**Problem Statement:**
Let \( A \) be the linear transformation \( \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) that first reflects vectors across the line \( y = -x \), then rotates vectors 90 degrees counterclockwise about the origin. Find the matrix representation of \( A \).
**Solution:**
To find the matrix representation of \( A \), we need to consider the two transformations: reflection across the line \( y = -x \) and rotation by 90 degrees counterclockwise.
**Reflection across the line \( y = -x \):**
The reflection of a point \( (x, y) \) across the line \( y = -x \) can be represented by the matrix:
\[ R = \begin{pmatrix}
0 & -1 \\
-1 & 0
\end{pmatrix} \]
**Rotation by 90 degrees counterclockwise:**
The rotation of a point \( (x, y) \) by 90 degrees counterclockwise can be represented by the matrix:
\[ Q = \begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix} \]
**Combined Transformation:**
Since the transformation \( A \) first reflects vectors across the line \( y = -x \), and then rotates vectors 90 degrees counterclockwise, the combined transformation can be represented by the product of the rotation matrix \( Q \) and the reflection matrix \( R \):
\[ A = Q \cdot R \]
Calculating the product:
\[ A = \begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix}
\begin{pmatrix}
0 & -1 \\
-1 & 0
\end{pmatrix}
= \begin{pmatrix}
(-1 \cdot -1) + (0 \cdot 0) & (0 \cdot -1) + (-1 \cdot 0) \\
(1 \cdot -1) + (0 \cdot 0) & (0 \cdot -1) + (1 \cdot 0)
\end{pmatrix}
= \begin{pmatrix}
1 & 0 \\
-1 & 0
\end{pmatrix} \]
The matrix
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