# 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 $$\mathbb{R}$$?)

What are some good notes, textbooks or other sources to learn about this subject?

• 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.
Commented 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
Commented 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 $GF(p)$. Commented Dec 5, 2021 at 19:22

## 2 Answers

Strangely, linear algebra specific to finite fields is best studied in textbooks on the theory of error-correcting codes (for example, MacWilliams and Sloane).

Pretty much all familiar notions in linear algebra extend to finite fields and GF(2). The notable exceptions are: 1) Orthogonal space may have a nontrivial intersection with the original space. That can cause significant confusion. Over GF(2), it is even possible to have a linear space that is its own orthogonal space. 2) There is no effective counterpart to spectral decomposition over finite fields. (speaking of which, does anyone know about existing efforts to address this "shortcoming"?)

As you mention, standard Linear Algebra books demonstrate the notions using infinite fields (usually the reals and the complex numbers). Pretty much, though, all the definitions are general enough to work for finite fields, too. The only place I know where you can find linear algebra (mostly vector spaces) for finite fields is in the preliminaries of books that deal with applications of finite field linear algebra, like the notes you mention. Another example is coding theory, where a standard reference is "A first course in coding theory" by Raymond Hill. However, the "chapter" on vector spaces over finite fields is not more than 6 pages. A similar book would be "Coding Theory: A First Course" by San Ling and Chaoping Xing.