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)
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