Theoretical Computer Science
reference-request linear-algebra coding-theory finite-fields
Updated Mon, 20 Jun 2022 11:04:39 GMT

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

## Solution

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"?)