I am new to this forum and I have a doubt with linear algebra, specifically linear maps and vectors. I have a linear map $R_\alpha = \begin{pmatrix} cos \alpha & - sin \alpha\\ sin \alpha & cos \alpha\\ \end{pmatrix} $ and the question goes like, how does $R_\alpha$ transform a vector?
Can someone please help me? I am not sure if this sort of queston as been asked, I tried to google but I did not get any results regarding this problem.
Any help is appreciated.
The comments above on the OP are good, but here's another perspective. How does $R_\alpha$ change the length of a vector? It is easy to verify that
$$R_\alpha^T R_\alpha=\begin{pmatrix}\cos^2 \alpha+\sin^2 \alpha & 0 \\ 0 & \cos^2 \alpha+\sin^2 \alpha \end{pmatrix}=I$$
So that for some vector $v$ we have $||R_\alpha v||^2=(R_\alpha v)^T (R_\alpha v)=v^T R_\alpha^T R_\alpha v=v^T I v=v^T v=||v||^2$. That is, $R_\alpha$ is an isometry (it can't change the length of vectors). So if $R_\alpha$ cannot change the length of a vector, it could only possibly rotate a vector. So what is this angle? We can compute the angle, $\theta$, between $R_\alpha v$ and $v$ by computing
$$\cos \theta=\frac{(R_\alpha v) \cdot v}{||R_\alpha v||\cdot||v||}.$$
I'll leave it to you to verify that this gives $\theta=\alpha$ (mod $\pi$ or whatever).