# Thread: orthogonal complement

1. ## orthogonal complement

Let W = span $\displaystyle $\left \{ \begin{bmatrix} 1\\ 2\\ -1 \end{bmatrix}, \begin{bmatrix} -1\\ 3\\ 2 \end{bmatrix} \right \}$$

Find a basis for the $\displaystyle $W^{\perp }$$

I am really confused on how to find this.

So i turned it it a matrix and got the rref.
$\displaystyle $\begin{bmatrix} 1 & -1\\ 2 &3 \\ -1& 2 \end{bmatrix}\sim \begin{bmatrix} 1 & 0\\ 0 & 1\\ 0& 0 \end{bmatrix}$$

I must've did this wrong because the answer is $\displaystyle $\begin{bmatrix} 7/5 \\ -1/5 \\ 1 \end{bmatrix}$$

I understand how they got the 1, but I have no idea how they got 7/5 and -1/5.
Can someone please explain. Thanks in advance

2. First of all W^p have only one vector(why?)

Recall the definition of vectors in W^p...

Suppose (a,b,c) is in W^p then:

You will get:

(1,2-1)(a,b,c)=0
(-1,3,3)(a,b,c)=0

A system of two equations with three variables.