a) Find the standard matrix of the linear transformation $T$, if $T:R^2 \rightarrow R^2$ reflects points through the line $y=x$ and then rotates points counterclockwise through π/4 radians.
b) Find and draw the image of the triangle with vertices (2,1), (1, 2), (2,2)
I think I can do the transformation, but I get confused easily by terminology. Do I literally just draw a triangle given the points, or is their more to it? If so, how do I solve b?
What is the answer to a? I would like to compare answers. I'm getting [[0, sin(π/4)][-sin(π/4), 0]].
Best Answer
At Julian's request, I expand my comment of 10 December 2012 into an answer.
$(1,0)$, reflected in $y=x$, becomes $(0,1)$; then rotating ccl through $\pi/4$ takes $(0,1)$ to $v=(-\sqrt2/2,\sqrt2/2)$.
$(0,1)$ reflects to $(1,0)$, which rotates to $w=(\sqrt2/2,\sqrt2/2)$.
The matrix we are looking for is the one whose columns are $v$ and $w$, that is, $$A=\pmatrix{-\sqrt2/2&\sqrt2/2\cr\sqrt2/2&\sqrt2/2\cr}$$
The image of the triangle with vertices $r=(2,1)$, $s=(1,2)$, and $t=(2,2)$ is the triangle whose vertices are $Ar$, $As$, and $At$. We leave it to the reader to carry out the multiplications.