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 . . .
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.