# Orthogonal projection

• Aug 9th 2010, 05:41 PM
wilday86
Orthogonal projection
Find the projwv for the given vector v and the subspace W.

Let V be the Euclidean space R^4 and W the subspace with the basis
$$\begin{bmatrix} 1 & 1 &0 &1 \end{bmatrix},\begin{bmatrix} 0 & 1 &1 &0 \end{bmatrix},\begin{bmatrix} -1 & 0 &0 &1 \end{bmatrix}$
$

$$v=\begin{bmatrix} 2 & 1 &3 &0 \end{bmatrix}$
$

My work:
$$proj_{w}v= \frac{}{}w_{1}+\frac{}{}w_{2}+\frac{}{}w_{3}$
$

$$= \frac{\begin{bmatrix} 2\\ 1\\ 3\\ 0 \end{bmatrix}\cdot\begin{bmatrix} 1\\ 1\\ 0\\ 1 \end{bmatrix}}{\begin{bmatrix} 1\\ 1\\ 0\\ 1 \end{bmatrix}\cdot\begin{bmatrix} 1\\ 1\\ 0\\ 1 \end{bmatrix}}\begin{bmatrix} 1\\ 1\\ 0\\ 1 \end{bmatrix}+\frac{\begin{bmatrix} 2\\ 1\\ 3\\ 0 \end{bmatrix}\cdot\begin{bmatrix} 0\\ 1\\ 1\\ 0 \end{bmatrix}}{\begin{bmatrix} 0\\ 1\\ 1\\ 0 \end{bmatrix}\cdot\begin{bmatrix} 0\\ 1\\ 1\\ 0 \end{bmatrix}}\begin{bmatrix} 0\\ 1\\ 1\\ 0 \end{bmatrix}+\frac{\begin{bmatrix} 2\\ 1\\ 3\\ 0 \end{bmatrix}\cdot\begin{bmatrix} -1\\ 0\\ 0\\ 1 \end{bmatrix}}{\begin{bmatrix} -1\\ 0\\ 0\\ 1 \end{bmatrix}\cdot\begin{bmatrix} -1\\ 0\\ 0\\ 1 \end{bmatrix}}\begin{bmatrix} -1\\ 0\\ 0\\ 1 \end{bmatrix}$
$

$$= \frac{3}{3}\begin{bmatrix} 1\\ 1\\ 0\\ 1 \end{bmatrix}+\frac{4}{2}\begin{bmatrix} 0\\ 1\\ 1\\ 0 \end{bmatrix}+\frac{-2}{2}\begin{bmatrix} -1\\ 0\\ 0\\ 1 \end{bmatrix}= \begin{bmatrix} 1\\ 1\\ 0\\ 1 \end{bmatrix}+\begin{bmatrix} 0\\ 2\\ 2\\ 0 \end{bmatrix}+\begin{bmatrix} 1\\ 0\\ 0\\ -1 \end{bmatrix}=$
$

I stopped there because my result was no where near the correct answer :-/
The correct answer is $$\begin{bmatrix} 7/5 &11/5 & 9/5 &-3/5 \end{bmatrix}$
$

Can someone please tell me what I did wrong. Thanks in advance.
• Aug 9th 2010, 06:40 PM
TKHunny