<s>[INST] Find the coordinates for $v$ in the subspace $W$ spanned by the following vectors: 
I'm confused as to how I solve this. I was told to project each $u$ onto $v$ but how would that get me to the answer that's required?
 [/INST] Resolve the data vector into its components in the $u$ coordinate system. because the $u$ vectors are orthogonal, we can solve for the projection terms independently.

$$
\begin{array}{cccccccccc}
 v & = & 
\frac{\langle v,u_{1} \rangle}{\langle u_{1},u_{1} \rangle} & u_{1} & + & \frac{\langle v,u_{2} \rangle}{\langle u_{2},u_{2} \rangle} & u_{2} & + &
\frac{\langle v,u_{3} \rangle}{\langle u_{3},u_{3} \rangle} & u_{3} & \\
%
\left[
\begin{array}{r}
 5 \\ -2 \\ 9 \\ -5
\end{array}
\right]
& = &
 \frac{27}{22}  & \left[
\begin{array}{r}
 3 \\3 \\ 2 \\ 0
\end{array}
\right]
 & + & 
-\frac{11}{7}   & \left[
\begin{array}{r}
 2 \\ 0 \\ -3 \\ 1
\end{array}
\right]
 & + &
-\frac{65}{488} & \left[
\begin{array}{r}
 -6 \\ 2 \\ 6 \\ 30
\end{array}
\right] \\
%
&=&
\frac{1}{18788 }  & \left[
\begin{array}{r}
 -68799 \\ 101745 \\ -49419 \\ -10659
\end{array}
\right]
%
\end{array}
$$
</s>