<s>[INST] Rotate the standard basis and determine the bases that arise from it 
Given is $\mathbb{R}^2$ with standard basis $B^2_0$
A "new" basis $B=\left\{\vec{b_1}; \vec{b_2}\right\}$ arises from
  $B^2_0$ by rotation of $30°$ (clockwise). Determine the change of
  basis $T^{B}_{B^2_0}$ and the basis vectors $\vec{b_1}$ and
  $\vec{b_2}$.

So we are in $\mathbb{R}^2$ and we have a standard basis $B^2_0$. Standard matrix is another word for unit matrix I think, so we have 
$B^2_0 =\begin{pmatrix}
1 & 0\\ 
0 & 1
\end{pmatrix}$. 
Rotate this by $30°$, we have $B=
\begin{pmatrix}
\cos(30°) & -\sin(30°)\\ 
\sin(30°) &  \cos(30°)
\end{pmatrix} = 
\begin{pmatrix}
\frac{\sqrt{3}}{2} & -\frac{1}{2}\\ 
\frac{1}{2} & \frac{\sqrt{3}}{2}
\end{pmatrix}$
So $\vec{b_1} = \begin{pmatrix}
\frac{\sqrt{3}}{2}\\ 
\frac{1}{2}
\end{pmatrix}$ and $\vec{b_2} = 
\begin{pmatrix}
-\frac{1}{2}\\ 
\frac{\sqrt{3}}{2}
\end{pmatrix}$
And $T^{B}_{B^2_0}$ we can solve with Gaussian elimination, right? But I'm more interested to know if I calculated the rotation matrix and the resulting bases correctly?
 [/INST] To calculate the rotation matrix, with reference to the standard basis, it suffices to consider what are the transformed vectors for $\vec e_1$ and $\vec e_2$ after the rotation, that is
$$\vec{e_1} = \begin{pmatrix}
1\\ 
0
\end{pmatrix}\to \vec{b_1} = \begin{pmatrix}
\frac{\sqrt{3}}{2}\\ 
\frac{1}{2}
\end{pmatrix}
$$
$$\vec{e_2} = \begin{pmatrix}
0\\ 
1
\end{pmatrix}\to \vec{b_2} = 
\begin{pmatrix}
-\frac{1}{2}\\ 
\frac{\sqrt{3}}{2}
\end{pmatrix}$$
</s>