## 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

\(\begin{pmatrix} \cos \varphi & -\sin \varphi \\ \sin \varphi & \cos \varphi \\ \end{pmatrix}\)