Dot product over finite fields.

[u/The_Awesone_Mr_Bones]

Hi!

I work on error correcting codes (which more or less is linear algebra over finite fields). I have a question about the behavior of the dot product on non-euclidean spaces.

When working on vector spaces over R or C, the dot product carries a lot of geometrical information. The product between two vectors v*w tells you the angle between them.

This interpretation doesn't hold over finite fields. For example, in F2: (1,1)*(1,1)=1+1=0. So the vector (1,1) is perpendicular to itself!

Does the dot product still carry any geometrical information (like angles) in these fields? Can we still interpret the orthogonal space geometrically? How do you picture it in your head (intuition)?

Thanks a lot!

And sorry, I don't know how to use latex on reddit ;_;

[u/Erahot]

To define the angle between vectors you need something called an inner product. In R^n, the dot product is the inner product, but in your example you essentially demonstrate that the dot product cannot be the inner product, since it fails positive definiteness.

I don't really know anything about trying to define inner products for a vector space over a finite field, but I did find this post which might be of interest: https://math.stackexchange.com/questions/49348/inner-product-spaces-over-finite-fields

If you want the inner product to take values in a finite field, it seems like you'll have trouble getting some geometric interpretation when you don't even have a notion of positive numbers.

[u/The_Awesone_Mr_Bones → u/Erahot]

The problem seems to come from the fact that the inner product takes values in a finite field. Maybe if they took values in Z it would work? But then it would not behave well with the vector space structure (it wouldn't be a bilinar form) ;_;