[Math] Examples of Linear Transformation problem

linear-transformations

Assume that T is a linear transformation. Find the standard matrix of T.
$T:\mathbb{R}^2 \rightarrow\mathbb{R}^2$
first reflects points through the line $x_2$=$x_1$ and then reflects points through the horizontal $x_1$-axis.

My Solution , that is incorrect :-
The standard matrix for the reflection through the line $x_2$=$x_1$ is
\begin{bmatrix}0&1\\1&0\end{bmatrix}

The standard matrix for the reflection through the horizontal $x_1$-axis is :
\begin{bmatrix}1&0\\0&-1\end{bmatrix}

When we multiply this we get :
\begin{bmatrix}0&-1\\1&0\end{bmatrix}

This answer is being rejected. Can you please advise me what am I doing wrong?
Thank you.

Best Answer

The task is simple . . .

Let $B$ be the matrix for the transformation which reflects a point about the line $x_2=x_1$. Then $$B = \begin{bmatrix}0&1\\1&0\end{bmatrix}$$

Let $A$ be the matrix for the transformation which reflects a point about the $x_1$-axis. Then $$A = \begin{bmatrix}1&0\\0&-1\end{bmatrix}$$ The matrix for the transformation which first reflects a point about the line $x_2=x_1$, and then reflects the result about the $x_1$-axis is just $$AB = \begin{bmatrix}1&0\\0&-1\end{bmatrix}\begin{bmatrix}0&1\\1&0\end{bmatrix}=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$$

To test it, let $v={\displaystyle{\begin{bmatrix}x_1\\x_2\end{bmatrix}}}$.

First apply the transformations by hand . . .

  • Start with $$\begin{bmatrix}x_1\\x_2\end{bmatrix}$$
  • Next, reflect about the line $x1=x2$: $$\begin{bmatrix}x_1\\x_2\end{bmatrix} \rightarrow \begin{bmatrix}x_2\\x_1\end{bmatrix}$$
  • Next, reflect the above result about the $x_1$-axis: $$\begin{bmatrix}x_2\\x_1\end{bmatrix} \rightarrow \begin{bmatrix}x_2\\-x_1\end{bmatrix}$$

Next, check to see if $(AB)v$ yields the same result: $$(AB)v = \begin{bmatrix}0&1\\-1&0\end{bmatrix} \begin{bmatrix}x_1\\x_2\end{bmatrix} = \begin{bmatrix}x_2\\-x_1\end{bmatrix}$$ so it checks.

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