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Linear Algebra

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Moderate

Rank-Nullity of Orthogonal Projection

LINALG-J19FQ4

If $W\leq\mathbb{R}^8$ where $W$ has orthogonal basis $\{\vec{w}_1,\vec{w}_2,\vec{w}_3\}$ then the orthogonal projection (with respect to $W$) of a vector $\vec{v}\in\mathbb{R}^8$ is defined as:

$$proj_{W}(\vec{v})=\frac{\vec{v}\vec{w}_1}{\vec{w}_1\vec{w}_1}\vec{w}_1+\frac{\vec{v}\vec{w}_2}{\vec{w}_2\vec{w}_2}\vec{w}_2+\frac{\vec{v}\vec{w}_3}{\vec{w}_3\vec{w}_3}\vec{w}_3$$

...which makes $proj_{W}:\mathbb{R}^8\rightarrow W$ a function. Determine $nullity(proj_{W})$ and $rank(proj_{W})$.

A

$nullity=3$ and $rank=5$

B

$nullity=5$ and $rank=3$

C

$nullity=6$ and $rank=2$

D

$nullity=2$ and $rank=6$

E

These are not defined since $proj_{W}$ is not a linear transformation.