Assume that $T$ is linear transformation. Find the matrix of $T$.

a) $T: R^2 $ → $ R^2 $ first rotates points through $ -\frac {3π}{4} $ radians (clockwise) and then reflects points through the horizontal $x_1$-axis.

b) $T: R^2 $ → $ R^2 $ first reflects points through the horizontal $x_1$-axis and then reflects points through the line $x_1=x_2$. Show that this transformation is merely a rotation about the origin. What is the angle of the rotation?

I am unsure where to start with this since i am new with this, so could anyone explain to me how to solve this? Thanks!

  • $\begingroup$ Hint: For linear transformations, the matrix of the composition is the product of the matrices $\endgroup$ – Danilo Gregorin Sep 23 '17 at 20:34

I'll do a and let you tackle the second. A rotation matrix in $\mathbb{R}^2$ representing rotation by angle $\theta$ is given by $$ \begin{bmatrix}\cos \theta&-\sin \theta\\ \sin\theta&\cos\theta\end{bmatrix} $$ reflection accross the $x$ axis is given by $$ \begin{bmatrix}1&0\\ 0&-1\end{bmatrix} $$ so in your case we have $$ \begin{bmatrix}1&0\\ 0&-1\end{bmatrix}\begin{bmatrix}\cos -\frac{3\pi}{4}&-\sin -\frac{3\pi}{4}\\ \sin-\frac{3\pi}{4}&\cos-\frac{3\pi}{4}\end{bmatrix}= \begin{bmatrix}1&0\\ 0&-1\end{bmatrix}\begin{bmatrix}-\frac{\sqrt{2}}{2}& -\frac{\sqrt{2}}{2}\\ \frac{\sqrt{2}}{2}&-\frac{\sqrt{2}}{2}\end{bmatrix} $$ carrying out the matrix multiplication yields $$ \begin{bmatrix}-\frac{\sqrt{2}}{2}& -\frac{\sqrt{2}}{2}\\ -\frac{\sqrt{2}}{2}&\frac{\sqrt{2}}{2}\end{bmatrix} $$

  • $\begingroup$ Is this clockwise or counter-clockwise rotation? (Since op mentions $-3\pi/4$ clockwise and I belive the rotation matrix you give is for counter-clockwise rotations). $\endgroup$ – Epiousios Sep 23 '17 at 21:53
  • $\begingroup$ @1524 nuts, I didn't catch that $\endgroup$ – qbert Sep 23 '17 at 22:02
  • $\begingroup$ One should stress the fact that it is the matrix of a symmetry (symmetry composed with rotation = symmetry). $\endgroup$ – Jean Marie Sep 23 '17 at 22:47
  • $\begingroup$ Regarding the direction of rotation. If you are in doubt, about how a matrix transforms your space, just look at what your matrix does to the principal component vectors $\pmatrix{1\\0},\pmatrix{0\\1}$ $\endgroup$ – Doug M Sep 23 '17 at 23:10

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