Rotation around an axis

#linear_algebra

Let $T : R^{3} \to R^{3}$ be the linear transformation that rotates counterclockwise around the $z$ -axis by $\frac{2 π}{3}$ .

Write the matrix for $T$ with respect to the standard basis ${[\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]}$ .
Write the matrix for $T$ with respect to the basis ${[\begin{matrix} \frac{\sqrt{3}}{2} \\ - \frac{1}{2} \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]}$ .
Determine all (complex) eigenvalues of $T$ .
Is $T$ diagonalizable over $C$ ? Justify your answer.

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Let $T:{\bf R}^3\to {\bf R}^3$ be the linear transformation that rotates counterclockwise around the $z$-axis by $\frac{2\pi}{3}$.
\begin{enumerate}[label=\alph*)]
	\item Write the matrix for $T$ with respect to the standard basis $\left\{\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix},\begin{bmatrix} 0 \\ 1 \\ 0\end{bmatrix},\begin{bmatrix} 0 \\ 0 \\ 1\end{bmatrix}\right\}$.
	\item Write the matrix for $T$ with respect to the basis $\left\{\begin{bmatrix} \frac{\sqrt{3}}{2} \\ -\frac{1}{2} \\ 0 \end{bmatrix},\begin{bmatrix} 0 \\ 1 \\ 0\end{bmatrix},\begin{bmatrix} 0 \\ 0 \\ 1\end{bmatrix}\right\}$.
	\item Determine all (complex) eigenvalues of $T$.
	\item Is $T$ diagonalizable over ${\bf C}$? Justify your answer.
\end{enumerate}