Problem:

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

$\displaystyle W = span\left \{ \begin{bmatrix}1\\ 2\\ 3\\4\end{bmatrix},\begin{bmatrix}5\\ 6\\ 7\\8\end{bmatrix} \right \}$

Attempt:

First I tried taking an arbitrary 4x1 matrix, and dotting it with a vector from the span of W. Then I realized that it must be orthogonal to all of the vectors in the span of W, so I trashed that idea.

I know that $\displaystyle W^{\perp}$ is the kernel of the orthogonal projection onto W. So therefore, $\displaystyle W^{\perp} = proj_{W}(\vec{x}) = (\vec{u}_{1} \cdot \vec{x})\vec{u}_{m}+...+(\vec{u}_{m} \cdot \vec{x})\vec{u}_{m}$

This is where I get lost, and I'm not sure how to continue or even if I'm on the right track. Any help is greatly appreciated.

Here is my work: