I'm pretty confused on this question. How would I find the standard matrix for a linear transformation like this? I thought since it is mapping a point from (x,y) -> (x,-x) the solution should have been {(1,0) , ( -1,0)} but apparently that is wrong. Any clarification would be really appreciated
Best Answer
Reflection through the X-axis is $$(x,y) \mapsto (x,-y)$$ Reflection through the line $y=-x$ is $$(x',y') \mapsto (-y',-x')$$(First drop a perpendicular from $(x',y')$ to the line $y=-x$. Call the foot $(x_*,y_*)$ Call the reflected point$(x_{**},y_{**})$Use $$x_{**}-x_*=x_*-x',y_{**}-y_*=y_*-y'$$Conclude that $$x_{**}=-y',y_{**}=-x')$$ The composite transformation is given by $$\begin{bmatrix}x''\\y''\end{bmatrix}=\begin{bmatrix}0&-1\\-1&0\end{bmatrix}\begin{bmatrix}1&0\\0&-1\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}$$ $$=\begin{bmatrix}0&1\\-1&0\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}$$