Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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.](https://content.bartleby.com/qna-images/question/d2a99bde-e99a-4ca5-bad3-91849c93fae5/05cd0a9d-89e3-4629-a873-635bd695a03e/a0dmwsc_processed.png)
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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