# Reference request for linear algebra over GF(2)

I have been looking for materials on the linear algebra over $$GF(2)$$ but so far I haven't found any substantial textbooks or notes on this subject. In fact in one of the notes I found the introduction states that,

Normally, we would cite a series of useful textbooks with background informa- tion but amazingly there is no text for finite field linear algebra. We do not know why this is the case.

In particular I would like to learn about elementary notions such as linear independence, orthogonality, linear equations, rank and kernels, etc. I would especially like to understand what the implications of this properties are when considering the same problem over $$\mathbb{R}$$. (Eg. does linear independence over $$GF(2)$$ imply linear independence over $$\mathbb{R}$$? Does orthogonality over $$GF(2)$$ imply linear independence over $$\mathbb{R}$$?)

• Any book on linear algebra applies to linear algebra over any field, and so applies to the field $GF(2)$ as a special case. If you have specific questions, e.g. , about the relation between independence over $GF(2)$ vs over $\mathbb{R}$, I think it would be better to ask them individually on Math.SE.
– D.W.
Nov 18, 2021 at 7:37
• Most books and notes I am aware of explicitly assume that the field is either the reals or the complex numbers. And I’ve never seen a disclaimer in any set of notes that eg. Gram Schmidt might not work, or that nonzero vectors might be self-orthogonal. @D.W.
– gen
Nov 18, 2021 at 14:02
• @D.W.: Suppose you have a vector space $V \in \mathbb{F}^n$ and you take its dual vector space $W = \{w|w\cdot v = 0 \ \forall v \in V\}$, then over the reals $V+W = \mathbb{F}^n$. This is not true over $GF(p)$. Dec 5, 2021 at 19:22