1.
\begin{equation*}
A=\begin{pmatrix}\cos\theta\amp -\sin\theta\\\sin\theta\amp\cos\theta\end{pmatrix}
\end{equation*}
Exercises 1.10 Exercises
Exercise Group.
Compute the kernel of the linear map corresponding to the given matrix (see Remark 1.7.7). Furthermore, find all eigenvalues in \(\R\) and \(\C\) and corresponding eigenvectors.
2.
\begin{equation*}
A=\begin{pmatrix}1\amp 1\\1\amp 1\end{pmatrix}
\end{equation*}
3.
\begin{equation*}
A=\begin{pmatrix}0\amp -1\amp 0\\-1\amp 0\amp 0\\0\amp 0\amp -1\end{pmatrix}
\end{equation*}
4.
\begin{equation*}
A=\begin{pmatrix}-1\amp 0\amp 0\\0\amp 0\amp -1\\0\amp 1\amp 0\end{pmatrix}
\end{equation*}
5.
\begin{equation*}
A=\begin{pmatrix}a\amp b\\0\amp c\end{pmatrix}
\end{equation*}
where, \(a,b,c\in\K\text{.}\)
6.
\begin{equation*}
A=\begin{pmatrix}a\amp 0\\0\amp b\end{pmatrix}
\end{equation*}
where, \(a,b\in\K\text{.}\)
7.
\begin{equation*}
A=\begin{pmatrix}1\amp i\\-i\amp 1\end{pmatrix}
\end{equation*}
8.
\begin{equation*}
A=\begin{pmatrix}2\amp 0\amp 0\\1\amp 2\amp 0\\0\amp 1\amp 2\end{pmatrix}
\end{equation*}
9.
Consider the following matrix.
\begin{equation*}
A=\begin{pmatrix}-1\amp 0\\2\amp 1\end{pmatrix}
\end{equation*}
Find \(A^{10}\begin{pmatrix}2\\0\end{pmatrix}\text{.}\)
10.
Find a matrix \(A\in M_2(\R)\) such that \(A^2=I_2\text{,}\) \(A\neq I_2\text{,}\) and \(A\neq -I_2\text{.}\) List all its eigenvalues in \(\R\) and find corresponding eigenvectors. Describe action of the linear map corresponding to \(A\) geometrically.
