## Projection to the subspace spanned by a vector

## Problem 60

Let $T: \R^3 \to \R^3$ be the linear transformation given by orthogonal projection to the line spanned by $\begin{bmatrix}

1 \\

2 \\

2

\end{bmatrix}$.

**(a)** Find a formula for $T(\mathbf{x})$ for $\mathbf{x}\in \R^3$.

**(b)** Find a basis for the image subspace of $T$.

**(c)** Find a basis for the kernel subspace of $T$.

**(d)** Find the $3 \times 3$ matrix for $T$ with respect to the standard basis for $\R^3$.

**(e)** Find a basis for the orthogonal complement of the kernel of $T$. (The orthogonal complement is the subspace of all vectors perpendicular to a given subspace, in this case, the kernel.)

**(f)** Find a basis for the orthogonal complement of the image of $T$.

**(g)** What is the rank of $T$?

(*Johns Hopkins University Exam*)