I am working with the finite field $F_2$ which contains the elements $(0,1)$. The $+$ law is then done modulo $2$.
$F_2^n$ can be seen as an $F_2$ vector space. The problem I am facing is that I learnt linear algebra over $\mathbb{R}$ or $\mathbb{C}$. And I don't know what is still true when dealing with finite fields. A typical issue I faced was with orthogonality. Here the example:
We can define orthogonality properties based on the dot product, for instance for $n=3$, we have: $(010).(111)=0+1+0=1$
This is typically done in coding theory. Now the thing that confuses me a lot is that this dot product will not verify the standard axiom of an inner product, basically the positive condition. For instance, we can have the "weird" case in which $x.x = 0$ when $x \neq 0$. An example would be $(11).(11)=1+1=0$. Vectors that are self orthogonal exists !
Because of that, all my intuitions breaks down about what is true or not when dealing with finite fields.
I am thus looking for a nice book, dealing with the basics of linear algebra that works with finite fields (not only over $\mathbb{R}$ or $\mathbb{C}$). My main concern is that I don't have time to go in full details of such object. I really just need to get the basic intuition of what changes with respect to $\mathbb{R}$ or $\mathbb{C}$. And in practice I will only need to work with $F_2^n$.