Why this matrix is a rotation matrix and how it work? So I have the the column matrix $X=\begin{pmatrix}
a\\ 
b
\end{pmatrix}$ with $a,b$ real numbers and to this matrix I have the following vector associated: $\vec{x}=a\cdot \vec{i}+b\cdot\vec{j}$
I have the matrix: $A(t)=\begin{pmatrix}
\cos(t) & \sin(t)\\ 
-\sin(t) & \cos(t) 
\end{pmatrix}$ with $t$ real number
$B=\begin{pmatrix}
1\\ 
1
\end{pmatrix}$ , $U=A(-\frac{\pi}{12})\cdot B$ and $V=A(\frac{\pi}{6})\cdot B$
I need to find the cosinus of vectors $\vec{u}$ and $\vec{v}$ these vectors being associated to column matrix $U$ and $V$.The response is $\frac{\sqrt{2}}{2}$
So I just did calculations I it's a little bit of work but I find that the matrix A(t) is a rotation matrix with angle t and the cosinus between u and v is pi/4 but at school I didn't learn about rotation matrix.Can you explain me why the angle is pi/4? I mean, if you can, in simple terms how rotation matrix work.
Thanks!
 A: The important thing is that every rotation $\varphi$ about the origin is a linear transformation, meaning that, for all vectors $v, w$ of the plane and scalar $\lambda$, we have 
$$\varphi(\lambda v) =\lambda\cdot\varphi(v) \\
\varphi(v+w) =\varphi(v) +\varphi(w)$$
These properties can be easily seen geometrically, in particular, drawing the second one says something like '$\varphi$ takes parallelograms to parallelograms'.
It is also clear that the composition of rotation by angle $t$ with rotation by angle $s$ is the rotation by $s+t$.
Now, by the basics of linear algebra, since $\varphi$ is linear, once a basis $e_1,e_2$ is fixed, it can be expressed as multiplication by the matrix whose columns are $\varphi(e_1),\varphi(e_2)$ (coordinated in basis $e_1,e_2$). 
Consequently, composition of linear transformations corresponds to matrix multiplication.
In the given example $A(t)$ is the matrix of rotation by angle $-t$, with respect to the standard basis $i,j$. 
Based on the above, it should be straightforward that 
$$A(t)\cdot A(s)\ =\ A(t+s)$$
which boils down to the trigonometric addition formulas for $\sin$ and $\cos$ (and proves them at once). 
