 Marenonigt

2021-12-17

Assume that T is a linear transformation. Find the standard matrix of T.
$T:{R}^{2}⇒{R}^{2}$ first reflects points through the vertical ${x}_{2}-a\xi s$ and then reflects points through the line ${X}_{2}={X}_{1}$.
$A=?$ Jonathan Burroughs

Step 1
It is given that T is linear transformation
$T:{R}^{2}⇒{R}^{2}$
The standard matrix of T which firstly reflects the points through vertical ${x}_{2}-a\xi s$ and then reflects points through the line ${x}_{2}={x}_{1}$.
Let us consider the standard basis of ${R}^{2}$ that is
$\left\{{e}_{1}=\left(1,0\right),{e}_{2}=\left(0,1\right)\right\}$
Step 2
Reflection of point along vertical ${x}_{2}-a\xi s$.
For any
$\left({x}_{1},{x}_{2}\right)\in {R}^{2}$
$T\left({x}_{1},{x}_{2}\right)=\left(-{x}_{1},{x}_{2}\right)$
Firstly finding the images of standard basis under this transformation.
$T\left(1,0\right)=\left(-1,0\right)$
$T\left(0,1\right)=\left(0,1\right)$
Step 3
Reflection points through the line ${x}_{2}={x}_{1}$.
For any $\left({x}_{1},{x}_{2}\right)\in {R}^{2}$
$Y\left({x}_{1},{x}_{2}\right)=\left({x}_{2},{x}_{1}\right)$
Now finding the images of obtained points under the reflection of points along ${x}_{2}={x}_{1}$.
$T\left(-1,0\right)=\left(0,-1\right)$
$T\left(0,1\right)=\left(1,0\right)$
Therefore, the matrix of required linear transformation is given by
$A=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$ Fasaniu

Step 1
We are looking for a $2×2$ matrix that reflects points through the horizontal axis and then reflects points through the line ${x}_{2}={x}_{1}$.
First the points need to be reflected through the horizontal ${x}_{1}-a\xi s$. This then means that the ${x}_{1}-coordinate$ remains unaffected and the ${x}_{2}-coordinate$ changes sign:
${R}_{1}=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$
Note: if you multiply this matrix by the vector ${\left({x}_{1},{x}_{2}\right)}^{T}$ to the right, then you obtain the vector ${\left({x}_{1},-{x}_{2}\right)}^{T}$.
Step 2
Thus, the points need to be reflected through the line ${x}_{2}={x}_{1}$. The ${x}_{1}-coordinate$ and ${x}_{2}-coordinate$ then interchange.
${R}_{2}=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$
Note: if you multiply this matrix by the vector ${\left({x}_{1},{x}_{2}\right)}^{T}$ to the right, then you obtain the vector ${\left({x}_{2},{x}_{1}\right)}^{T}$.
Step 3
The result of the reflection matrices, with the first reflection matrix on the right, is then the combination of the reflections.
$T={R}_{2}×{R}_{1}=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]×\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$
Note: if you multiply this matrix by the vector ${\left({x}_{1},{x}_{2}\right)}^{T}$ to the right then you obtain the vector ${\left(-{x}_{2},{x}_{1}\right)}^{T}$

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