Let W be the subspace spanned by the uʹs. Write y as the sum of a vector in W and a vector orthogonal to W.
-$\mathbf { y } = \left[ \begin{array} { r } 17 \ 7 \ 12 \end{array} \right] , \mathbf { u } _ { 1 } = \left[ \begin{array} { r } 2 \ 2 \ - 1 \end{array} \right] , \mathbf { u } _ { 2 } = \left[ \begin{array} { r } - 1 \ 3 \ 4 \end{array} \right]$
A)
$$\mathbf { y } = \left[ \begin{array} { r }
6 \
14 \
4
\end{array} \right] + \left[ \begin{array} { r }
11 \
- 7 \
8
\end{array} \right]$$
B) $y=\left[\begin{array}{r}
6 \
14 \
4
\end{array}\right]+\left[\begin{array}{l}
23 \
21 \
16
\end{array}\right]$
C)
$\mathbf{y}=\left[\begin{array}{r}
12 \
28 \
8
\end{array}\right]+\left[\begin{array}{r}
5 \
-21 \
4
\end{array}\right]$

D)
$\mathbf{y}=\left[\begin{array}{r}
6 \
14 \
4
\end{array}\right]+\left[\begin{array}{r}
-11 \
7 \
-8
\end{array}\right]$

Question

Let W be the subspace spanned by the uʹs. Write y as the sum of a vector in W and a vector orthogonal to W.
-$\mathbf { y } = \left[ \begin{array} { r } 17 \ 7 \ 12 \end{array} \right] , \mathbf { u } _ { 1 } = \left[ \begin{array} { r } 2 \ 2 \ - 1 \end{array} \right] , \mathbf { u } _ { 2 } = \left[ \begin{array} { r } - 1 \ 3 \ 4 \end{array} \right]$
A)
$$\mathbf { y } = \left[ \begin{array} { r } 
6 \
14 \
4
\end{array} \right] + \left[ \begin{array} { r } 
11 \
- 7 \
8
\end{array} \right]$$
B) $y=\left[\begin{array}{r}
6 \
14 \
4
\end{array}\right]+\left[\begin{array}{l}
23 \
21 \
16
\end{array}\right]$
C)
$\mathbf{y}=\left[\begin{array}{r}
12 \
28 \
8
\end{array}\right]+\left[\begin{array}{r}
5 \
-21 \
4
\end{array}\right]$

D)
$\mathbf{y}=\left[\begin{array}{r}
6 \
14 \
4
\end{array}\right]+\left[\begin{array}{r}
-11 \
7 \
-8
\end{array}\right]$

Quizplus · Accepted Answer

To write $\mathbf{y}$ as the sum of a vector in $W$ and a vector orthogonal to $W$, we first find the projection of $\mathbf{y}$ onto $W$, which is spanned by $\mathbf{u}_1$ and $\mathbf{u}_2$. The projection formula for a vector $\mathbf{v}$ onto a subspace spanned by vectors $\mathbf{u}_1$ and $\mathbf{u}_2$ is given by:$$\text{proj}_W \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{u}_1}{\mathbf{u}_1 \cdot \mathbf{u}_1}\mathbf{u}_1 + \frac{\mathbf{v} \cdot \mathbf{u}_2}{\mathbf{u}_2 \cdot \mathbf{u}_2}\mathbf{u}_2$$After finding the projection, the vector in $W$ is the projection itself, and the vector orthogonal to $W$ is the difference between $\mathbf{y}$ and its projection onto $W$. The correct computation leads to the vectors given in choice A.

Let W Be the Subspace Spanned by the Uʹs A) y=[6144]+[11−78]\mathbf { y } = \left[ \begin{array} { r } 6 \\14 \\4\end{array} \right] + \left[ \begin{array} { r } 11 \\- 7 \\8\end{array} \right]y=​6144​​+​11−78​​

Let W Be the Subspace Spanned by the Uʹs A) $\mathbf { y } = \left[ \begin{array} { r } 6 \\14 \\4\end{array} \right] + \left[ \begin{array} { r } 11 \\- 7 \\8\end{array} \right]$