Find the standard matrix for the linear transformation T: R² → R2 that reflects points about the origin.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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The problem statement is as follows:

"Find the standard matrix for the linear transformation \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) that reflects points about the origin."

Below the statement, there is a matrix format with four empty boxes where the elements of the 2x2 matrix are to be filled in.

Additionally, there is a "Check Answer" button below the matrix.

**Explanation of the Problem:**

To solve this problem, you are asked to determine the standard matrix for a transformation that reflects vectors about the origin in a 2-dimensional plane.

**Reflection About the Origin:**

The matrix for reflecting about the origin in \(\mathbb{R}^2\) is given by:

\[
\begin{bmatrix}
-1 & 0 \\
0 & -1
\end{bmatrix}
\]

This matrix, when applied to any vector \(\mathbf{v} = \begin{bmatrix} x \\ y \end{bmatrix}\), transforms it to \(\begin{bmatrix} -x \\ -y \end{bmatrix}\), effectively reflecting it across the origin.
Transcribed Image Text:The problem statement is as follows: "Find the standard matrix for the linear transformation \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) that reflects points about the origin." Below the statement, there is a matrix format with four empty boxes where the elements of the 2x2 matrix are to be filled in. Additionally, there is a "Check Answer" button below the matrix. **Explanation of the Problem:** To solve this problem, you are asked to determine the standard matrix for a transformation that reflects vectors about the origin in a 2-dimensional plane. **Reflection About the Origin:** The matrix for reflecting about the origin in \(\mathbb{R}^2\) is given by: \[ \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \] This matrix, when applied to any vector \(\mathbf{v} = \begin{bmatrix} x \\ y \end{bmatrix}\), transforms it to \(\begin{bmatrix} -x \\ -y \end{bmatrix}\), effectively reflecting it across the origin.
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