
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

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:
(1,21)(a,b,c)=0
(1,3,3)(a,b,c)=0
A system of two equations with three variables.