Orthogonal projection and scaling

#linear_algebra

Let $L$ be the line in $R^{2}$ defined by $y = - 3 x$ , and let $T : R^{2} \to R^{2}$ be the linear transformation that orthogonally projects onto $L$ and then stretches along $L$ by a factor of two.

Find the eigenvalues and an eigenbasis $B$ for $T$ .
Determine the matrix for $T$ with respect to the basis $B$ .
Determine the matrix for $T$ with respect to the standard basis.

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Let $L$ be the line in ${\bf R}^2$ defined by $y=-3x$, and let $T:{\bf R}^2\to {\bf R}^2$ be the linear transformation that orthogonally projects onto $L$ and then stretches along $L$ by a factor of two.
\begin{enumerate}[label=\alph*)]
	\item Find the eigenvalues and an eigenbasis $\mathcal{B}$ for $T$.
	\item Determine the matrix for $T$ with respect to the basis $\mathcal{B}$.
	\item Determine the matrix for $T$ with respect to the standard basis.
\end{enumerate}