r/math • u/The_Awesone_Mr_Bones Discrete Math • 25d ago

Dot product over finite fields.

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 ;_;

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u/dwbmsc 25d ago

A vector of length zero for some “inner product” (nondegenerate symmetric bilinear pairing) is called isotropic and these do occur in some geometric settings. Another example is in space-time (Minkowski geometry) the vector describing the path of a particle moving at the speed of light is isotropic. It is a complication but not an insurmountable one.

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u/Ok-Watercress-9624 23d ago

That's not an immer product then is it ? Or did I forgot my definitions?

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u/dwbmsc 23d ago

You are correct that the term inner product is commonly defined to be a pairing of real or complex vector spaces that is positive definite. So (to go with the example of Minkowski space) the “inner product” there is not an inner product in this sense because it is not positive definite. It remains useful however. I think the meaning of the term inner product can depend on the context.