I have more a systems background, but I have a math-y type question so I figured I'd give it a shot here...This is more of an implementation question, I don't need to prove anything at the moment.
I'm trying to find the basis of the null space over $\mathbb{F}_2^N$ for a given basis quickly and I was hoping someone here would know how.
For example if I have the following basis in $\mathbb{F}_2^{16}$:
$$
\left(
\begin{array}{cccccccccccccccc}
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\
0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \\
0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\
\end{array}
\right)
$$
How would I find the null space basis for this matrix?
If I put my basis into reduced row echelon form, I could find it easily, but for my particular problem I cannot do that. I know there are exhaustive search methods, but the matrices I'm dealing with can be quite large, which make those impractical.
BEGIN EDIT
@Neil de Beaudrap, It has to do with the fact that I am actually splitting up the vector space and using part of it for another purpose. If I change this matrix with elementary row operations and put it into reduced-row-echelon form it messes things up....
I am unfamiliar with column operations, could you explain in a bit more detail what you are talking about? Thanks!
END EDIT
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