Orthogonal projection onto a line (3)

#linear_algebra

Let $L$ be the line $L$ parameterized by $L (t) = (2 t, - 3 t, t)$ for $t \in R$ , and let $T : R^{3} \to R^{3}$ be the linear transformation that is orthogonal projection onto $L$ .

Describe $\ker (T)$ and $im (T)$ , either implicitly (using equations in $x, y, z$ ) or parametrically.
List the eigenvalues of $T$ and their geometric multiplicities.
Find a basis for each eigenspace of $T$ .
Let $A$ be the matrix for $T$ with respect to the standard basis. Find a diagonal matrix $B$ and an invertible matrix $S$ such that $B = S^{- 1} A S$ . (You do not have to compute $A$ .)

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Let $L$ be the line $L$ parameterized by $L(t)=(2t,-3t,t)$ for $t\in {\bf R}$, and let $T:{\bf R}^3\to {\bf R}^3$ be the linear transformation that is orthogonal projection onto $L$.
\begin{enumerate}[label=\alph*)]
	\item Describe $\operatorname{ker}(T)$ and $\operatorname{im}(T)$, either implicitly (using equations in $x,y,z$) or parametrically.
	\item List the eigenvalues of $T$ and their geometric multiplicities.
	\item Find a basis for each eigenspace of $T$.
	\item Let $A$ be the matrix for $T$ with respect to the standard basis. Find a diagonal matrix $B$ and an invertible matrix $S$ such that $B=S^{-1}AS$. (You do not have to compute $A$.)
\end{enumerate}