Let \(T:\RR ^2\rightarrow \RR ^2\) be a linear transformation induced by some matrix \(A\). The sketch below shows two lines. Suppose the following:
  • when transformation \(T\) is applied to vectors along the line \(y=2x\), the vectors reverse direction, but keep their lengths.
  • when transformation \(T\) is applied to vectors along the line \(y=-x\), the vectors keep their direction, but triple in length.

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Find the eigenvalues of \(A\).

\[\text {Eigenvalues of A (in increasing order): }\lambda _1=\answer {-1}, \quad \lambda _2=\answer {3}\]

Find a basis for the eigenspace associated with each of these eigenvalues.

A basis for \(\mathcal {S}_{\lambda _1}\): \(\left \{\begin{bmatrix}1\\\answer {2}\end{bmatrix}\right \}\)

A basis for \(\mathcal {S}_{\lambda _2}\): \(\left \{\begin{bmatrix}1\\\answer {-1}\end{bmatrix}\right \}\)