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?
$\endgroup$ 42 Answers
$\begingroup$Scaling one axis but not the other will distort angles.
(Also, your result is obviously wrong, since you didn't get an orthogonal matrix.)
$\endgroup$ $\begingroup$i don't know why you are making it more difficult. you can use the projection of the unit vectors $(1,0)$ and $(0,1)$ onto the $(1,2).$
the image of $x$ onto $a$ is $a' = 2(a^Tx)a/(a^Ta) - x.$ using this your problem, for the image of $x = (1,0), a = (1,2).$ so we have $a^tx = 1, a^T a = 5$ which gives $a' = 2/5(1,2) - (1,0) = (-3/5, 2/5)$ this $a'$ is your first column of the reflection matrix. to find the second column find $(0,1)'.$
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