I've been doing some work in Linear Algebra for my course at school. I just want to be clear about how to find the orthogonal complement of a subspace. The basis for the subspace, W, is shown below, composed of 3 vectors:$$W = \begin{Bmatrix}\begin{bmatrix}1\\2\\3\\4 \end{bmatrix} \begin{bmatrix}-3\\4\\2\\6\end{bmatrix} \begin{bmatrix}2\\-2\\3\\5\end{bmatrix}\end{Bmatrix}$$
I would like to know if one simply sets $W*W^T$$=0$ and takes the columns of the resulting matrix as the basis of the orthogonal complement of W, provided that row reduction has been performed to make sure the remaining columns are linearly independent.
Please note that I have created an arbitrary set of vectors above that are not orthogonal, and so if they need to be for $W*W^T$$=0$ please point that out. Thank you.
Best Answer
Think of a system of linear equations you want to solve that will say your vector $\mathbf x$ is orthogonal to $W$.