# Find matrix representation of transformation Given two lines l 1 </msub> : y =

Find matrix representation of transformation
Given two lines ${l}_{1}:y=x-3$ and ${l}_{2}:x=1$ find matrix representation of transformation $f$(in standard base) which switch lines each others and find all invariant lines of $f$.
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eurgylchnj
Since the two straight lines intersect at the point $P=\left(1,-2\right)$ the transformation must have a fixed point in $P$. We can find such transformations in three steps.
1) translate the origin in $P$ with the translation ${T}_{P}^{-1}\left(x,y\right)\to \left(x-1,y+2\right)$
2) perform a simmetry $S$ with axis the strignt line passing thorough the new origin ad such that bisect the angle between the two lines.
3) return to the old origin with the translation ${T}_{P}\left(x,y\right)\to \left(x+1,y-2\right)$.
So the searched matrix has the form: $M={T}_{P}S{T}_{P}^{-1}$.
This is not a linear transformation but an affine one, and, if you want, can be represented by a $3×3$ matrix.
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icedagecs
The translation matrices in omogeneous coordinate are:
${T}_{P}=\left[\begin{array}{ccc}1& 0& -1\\ 0& 0& 2\\ 0& 0& 1\end{array}\right]\phantom{\rule{2em}{0ex}}{T}_{P}^{-1}=\left[\begin{array}{ccc}1& 0& 1\\ 0& 0& -2\\ 0& 0& 1\end{array}\right]$
and the reflection matrix can be found noting that the angle between the bisetrix and the $x$-axis is $\theta =\frac{3\pi }{8}$, Then the matrix is:
$S=\left[\begin{array}{ccc}\mathrm{cos}2\theta & \mathrm{sin}2\theta & 0\\ \mathrm{sin}2\theta & -\mathrm{cos}2\theta & 0\\ 0& 0& 1\end{array}\right]$
Note that the invariant lines are the bisector and his orthogonal in $P$.