# 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
$\displaystyle $\begin{bmatrix} 1 & 1 &0 &1 \end{bmatrix},\begin{bmatrix} 0 & 1 &1 &0 \end{bmatrix},\begin{bmatrix} -1 & 0 &0 &1 \end{bmatrix}$$

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

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

$\displaystyle $= \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}$$

$\displaystyle $= \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 $\displaystyle $\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