Geometric transforms in the plane
Task number: 2656
The following matrices represent a linear transformations in the plane \(\mathbb R^2\). Determine eigenvalues and the associated eigenvectors, and interpret these in geometric terms.
Variant 1
\(\begin{pmatrix} 2 & 0 \\ 0 & 2 \\ \end{pmatrix}\)
Variant 2
\(\begin{pmatrix} 2 & 0 \\ 0 & 1 \\ \end{pmatrix}\)
Variant 3
\(\begin{pmatrix} 2 & 1 \\ 0 & 2 \\ \end{pmatrix}\)
Variant 4
\(\begin{pmatrix} 0 & -1 \\ -1 & 0 \\ \end{pmatrix}\),
Variant 5
\({1 \over 2}\!\begin{pmatrix} 1 & 1 \\ 1 & 1 \\ \end{pmatrix}\)
Variant 6
\(\begin{pmatrix} 0 & -1 \\ 1 & 0 \\ \end{pmatrix}\)
Variant 7
\(\begin{pmatrix} \cos \varphi & -\sin \varphi \\ \sin \varphi & \cos \varphi \\ \end{pmatrix}\)
