Consider the linear transformation T: R<sup>2</sup> → R<sup>2</sup> that is defined by T(x) = Ax, where A is the
matrix
$$A = \frac{1}{17} \begin{pmatrix} 8 & 15 \\ 15 & -8 \end{pmatrix}$$
(a) Determine the eigenvalues λ<sub>1</sub> and λ<sub>2</sub> of A (where λ<sub>1</sub> < λ<sub>2</sub>).
(b) Determine an eigenvector v<sub>1</sub> for λ<sub>1</sub> and an eigenvector v<sub>2</sub> for λ<sub>2</sub>.
(c) Show that v<sub>1</sub> and v<sub>2</sub> are orthogonal.
(d) Show for all x ∈ R<sup>2</sup> that x is a linear combination of the eigenvectors of A.
(e) Evaluate T(x) in terms of v<sub>1</sub> and v<sub>2</sub>, and hence geometrically describe how T linearly
transforms R<sup>2</sup>.
Write a short description for each step.
