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.

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Chapter2: Second-order Linear Odes
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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
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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