Question:
Determine the matrix for reflection in the line 2𝑥 − 𝑦 = 0 in ℝ2
Hint: split this transformation into simpler ones, and combine the
result using matrix multiplication.
I thought about it this way.
y = 2x.
If I scale all y values down by 1/2 with the matrix,
\begin{pmatrix}
1 & 0 \\
0 & 1/2 \\
\end{pmatrix}
And do reflection as if y=x,
\begin{pmatrix}
0 & 1 \\
1 & 0 \\
\end{pmatrix}
And scale the y values back up by 2,
\begin{pmatrix}
1 & 0 \\
0 & 2 \\
\end{pmatrix}
If I multiply the matrices in the same order, I should get the reflection matrix for the line 2x-y=0
However, the result I had below was not correct.
\begin{pmatrix}
0 & 2 \\
1/2 & 0 \\
\end{pmatrix}
What is wrong about my approach?
Best Answer
Scaling one axis but not the other will distort angles.
(Also, your result is obviously wrong, since you didn't get an orthogonal matrix.)