Orthogonal projection onto a plane (2)

#linear_algebra

Let $a, b \in R$ and $T : R^{3} \to R^{3}$ be the linear transformation that is orthogonal projection onto the plane $z = a x + b y$ (with respect to the usual Euclidean inner-product on $R^{3}$ ).

Find the eigenvalues of $T$ and bases for the corresponding eigenspaces.
Is $T$ diagonalizable? Justify.
What is the characteristic polynomial of $T$ ?

View

L A T E X

code

Let $a, b \in {\bf R}$ and $T: {\bf R}^3 \to {\bf R}^3$ be the linear transformation that is orthogonal projection onto the plane $z=ax+by$ (with respect to the usual Euclidean inner-product on ${\bf R}^3$).
\begin{enumerate}[label=(\alph*)]
	\item Find the eigenvalues of $T$ and bases for the corresponding eigenspaces.
	\item Is $T$ diagonalizable? Justify.
	\item What is the characteristic polynomial of $T$?
\end{enumerate}