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I would like to ask for recommendations about texts that cover vector spaces over finite fields in detail.

I'm very interested in group theory, and one concept that seems to come up is vector spaces over finite fields as opposed to the usual infinite fields that tend to show up in undergraduate linear algebra classes ($\mathbb{R}$ and $\mathbb{C}$).

Now, I feel fairly comfortable with undergraduate level linear algebra, but I seem to have trouble intuitively understanding how things like linear independence and span would work with finite fields. For the linear algebra books I do have, they do not consider these. Are there any good sources on this?

Thank you.

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  • $\begingroup$ Well, most linear algebra books I know deal with vector spaces over arbitrary fields (thus including finite fields). What is different and interesting is automorphism groups (preserving structure). Take a look at Grove "Classical groups and geometric algebra" $\endgroup$ Commented Sep 9, 2021 at 17:29
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    $\begingroup$ This is a nice little book: amazon.com/Linear-Algebra-Geometry-Second-Mathematics/dp/… $\endgroup$
    – user637978
    Commented Sep 9, 2021 at 17:42
  • $\begingroup$ Thank you @BrauerSuzuki and Num Toez . I will be sure to look into these. $\endgroup$
    – Chris
    Commented Sep 9, 2021 at 17:44

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Now, I feel fairly comfortable with undergraduate level linear algebra, but I seem to have trouble intuitively understanding how things like linear independence and span would work with finite fields. For the linear algebra books I do have, they do not consider these.

That can’t be the case, because the definition of linear span and linear independence do not depend on the field being infinite. Nothing changes for finite fields in these cases, and all of basic linear algebra is going to hold for them.

The main thing that comes to mind that changes (compared to $\mathbb R, \mathbb C$ is using inner products. I wouldn’t call this basic linear algebra though: it would be basic vector geometry.

Now, a decent book on algebraic coding theory will explore vector spaces over all finite fields. They also discuss how inner products on them can be used for their purposes. The study of general bilinear forms over these fields probably leads to the connection to geometry alluded to in the comments.

I do not know exactly what you’re looking for, but it’s safe to say that this is the main distinction to keep an eye out for.

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