In answer to the question here:
Lights out game on hexagonal grid
The following argument is given:
"Thus if v is in the null space of A, then d is orthogonal to v and as a consequence, d is in the row space of A."
Here everything is over $\mathbb{F}_2$, and the "inner product" is the standard dot product $x\cdot y=\sum_{i=1}^nx_iy_i$.
However, this is not an inner product at all! It may very well be the case that $x\cdot x=0$ but $x\ne 0$.
I'm sure that a certain amount of linear algebra can still be preserved here. In particular, if I can prove that $\dim W + \dim W^\perp = \dim V$ it will suffice. So my main question here is how can I prove the above equality in this setting.