# Orthogonal projection onto span of vectors using weighted inner product

• Jul 14th 2008, 06:11 PM
JCS007
Orthogonal projection onto span of vectors using weighted inner product
Find the orthogonal projection of
$v=\begin{pmatrix}1 \\ 2 \\ -1 \\ 2 \end{pmatrix}$
onto the span of $\begin{pmatrix}1 \\ -1 \\ 2 \\ 5 \end{pmatrix}$ and $\begin{pmatrix}2 \\ 1 \\ 0 \\ -1 \end {pmatrix}$
using the weighted inner product $=4v_1w_1+3v_2w_2+2v_3w_3+v_4w_4$
• Jul 14th 2008, 08:43 PM
CaptainBlack
Quote:

Originally Posted by JCS007
Find the orthogonal projection of
$v=\begin{pmatrix}1 \\ 2 \\ -1 \\ 2 \end{pmatrix}$
onto the span of $\begin{pmatrix}1 \\ -1 \\ 2 \\ 5 \end{pmatrix}$ and $\begin{pmatrix}2 \\ 1 \\ 0 \\ -1 \end {pmatrix}$
using the weighted inner product $=4v_1w_1+3v_2w_2+2v_3w_3+v_4w_4$

Let:

$b_1=\begin{pmatrix}1 \\ -1 \\ 2 \\ 5 \end{pmatrix}$

and:

$b_2=\begin{pmatrix}2 \\ 1 \\ 0 \\ -1 \end {pmatrix}$

and:

$
u_1=\frac{b_1}{||b_1||}
$

$
u_2=\frac{b_2-\langle b_2,u_1 \rangle u_1}{||b_2-\langle b_2,u_1 \rangle u_1||}
$

Then the orthogonal projection is:

$p=\langle v,u_1\rangle u_1 + \langle v,u_2 \rangle u_2$

Where $||x||=\langle x,x \rangle^{\frac{1}{2}}$

RonL