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

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

Difficulty level: Easy task (using definitions and simple reasoning)
Routine calculation training
Cs translation
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