Find the closest point to y in the subspace W spanned by u 1 and u 2 .
-$$\mathbf { y } = \left[ \begin{array} { r }
27 \
17 \
7
\end{array} \right] , \mathbf { u } _ { 1 } = \left[ \begin{array} { r }
2 \
2 \
- 1
\end{array} \right] , \mathbf { u } _ { 2 } = \left[ \begin{array} { r }
- 1 \
3 \
4
\end{array} \right]$$
A)
$\left[ \begin{array} { c } - 8 \ 96 \ 95 \end{array} \right]$
B)
$\left[ \begin{array} { r } - 16 \ - 24 \ 1 \end{array} \right]$

C)
$\left[ \begin{array} { r } - 5 \ 31 \ 34 \end{array} \right]$

D)

$\left[ \begin{array} { c } 16 \ 24 \ - 1 \end{array} \right]$

Question

Find the closest point to y in the subspace W spanned by u 1 and u 2 .
-$$\mathbf { y } = \left[ \begin{array} { r } 
27 \
17 \
7
\end{array} \right] , \mathbf { u } _ { 1 } = \left[ \begin{array} { r } 
2 \
2 \
- 1
\end{array} \right] , \mathbf { u } _ { 2 } = \left[ \begin{array} { r } 
- 1 \
3 \
4
\end{array} \right]$$
A)
$\left[ \begin{array} { c } - 8 \ 96 \ 95 \end{array} \right]$
B)
$\left[ \begin{array} { r } - 16 \ - 24 \ 1 \end{array} \right]$

C)
$\left[ \begin{array} { r } - 5 \ 31 \ 34 \end{array} \right]$

D)

$\left[ \begin{array} { c } 16 \ 24 \ - 1 \end{array} \right]$

Quizplus · Accepted Answer

The closest point to y in the subspace W is the orthogonal projection of y onto W. The vectors u1 and u2 are orthogonal, so the projection of y onto W is given by the sum of the projections of y onto each of u1 and u2. Calculating these projections and summing them gives the vector in choice D.

Find the Closest Point to Y in the Subspace W