# Thread: Orthogonal Projection / Least Squares Problem

1. ## Orthogonal Projection / Least Squares Problem

Find the orthogonal projection of u = (5, 6, 7, 2) on the solution space of the homogeneous linear system:
$x_1 + x_2 + x_3 = 0$
$2x_2 + x_3 + x_4 = 0$

If I knew how to set this one up, I could run with it, but I'm a little confused. I know it will eventually involve the normal system $A^TAx = A^Tb$, but I'm not sure how to find A and b with a question worded this way.

2. Originally Posted by seuzy13
Find the orthogonal projection of u = (5, 6, 7, 2) on the solution space of the homogeneous linear system:
$x_1 + x_2 + x_3 = 0$
$2x_2 + x_3 + x_4 = 0$

If I knew how to set this one up, I could run with it, but I'm a little confused. I know it will eventually involve the normal system $A^TAx = A^Tb$, but I'm not sure how to find A and b with a question worded this way.
First you need to find a basis for the solution space of the linear system

$\begin{bmatrix}1 & 1 & 1& 0 \\ 0 & 2 & 1 & 1 \end{bmatrix} \begin{bmatrix}x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} = \begin{bmatrix}0 \\ 0 \end{bmatrix}$

The basis for the solution space is

$v_1=\begin{pmatrix} 1 \\ 1 \\ 0 \\ 2 \end{pmatrix}$ and $v_2=\begin{pmatrix} 1 \\ -1 \\ 2 \\ 0 \end{pmatrix}$

These are the columns of the Matrix $A$ the $b$ is the u given in your post. Now just use the formula you posted.

3. Okay, but I'm not sure I understand how you found the basis for the solution space of the system. =/