I am trying to calculate the null space of a binary matrix (GF(2)), but I don't want it to be orthonormal.
For instance, if you ask Wolfram|Alpha to calculate the null space of the following matrix: \begin{equation} A = \begin{bmatrix} 1 & 0 & 0 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 & 0 & 0 & 1 \end{bmatrix} \end{equation}
It outputs:
I'm trying to get the null space with Octave/MATLAB's function null(A). However, the result is what Wolfram|Alpha calls the Orthonormal basis, while what I want is just the Basis.
I know that $\sqrt{83} = \sqrt{(-3)^2 + (-2)^2 + ... + 5^2}$, but if I haven't got the values $(-3, -2, ...)$ I'm unable to reverse the process (or at least not without having to solve an equation system).
Is there any mathematical operation (or function similar to null) that I can use to transform it from orthonormal to non-orthonormal?