I am solving these problems in preparation to a midterm. Here's the problem I have solved before tackling the one I am asking help with:
I was given 1 basis vector for $\mathbf{W}$ - $\begin{bmatrix} 1\\ 2\\ 3\\ 4\\ 5 \end{bmatrix}$, and I had to find a basis for $\mathbf{W}^{\perp}$. I set up an equation $\begin{bmatrix} 1\\ 2\\ 3\\ 4\\ 5 \end{bmatrix}$ dotted with $\begin{bmatrix} a & b & c & d & e \end{bmatrix}$ = 0 and found the kernel (it was $\begin{bmatrix} -2b-3c-4d-5e\\ b\\ c\\ d\\ e \end{bmatrix}$.)
Now, I know that Span of $\mathbb{W}$ = $\begin{bmatrix} 1\\ 2\\ 3\\ 0 \end{bmatrix}$, $\begin{bmatrix} 1\\ 2\\ 3\\ 4 \end{bmatrix}$. I need to find a basis for $\mathbf{W}^{\perp}$, or the sub-space that contains vectors such that if two vectors from $\mathbf{W}$ and $\mathbf{W}^{\perp}$ were dotted, the result would be 0. But I am not sure how to go about this similar problem when I am given two vectors. Thanks!