hyperplanes in finite vector spaces

Question

how is an hyperplane in a finite vector space?

I know a hyperplane is the kernel of a linear map, and the dimension of the hyperplane is n-1 if dimension of the vector space is n.

So if I have, for example $\mathbb{Z}_2^3$ the vector space of all $(a,b,c)$ such that $a,b,c\in \mathbb{Z}_2$... how is an hyperplane here?

Thanks

Answer 1

A hyperplane in any vector space V is a subspace W such that V/W is one dimensional.

This is equivalent to defining them as kernels of linear maps into the base field.

It has nothing to do with the field or dimensionality whatsoever.

Answer 2

Consider the span $\lbrace (1,0,0),(0,1,0) \rbrace$. It is a two dimensional subspace of $\mathbf{Z_{2}}^{3}$ and is the kernel of the functional $T: \mathbf{Z_{2}}^{3} \to \mathbf{Z_{2}}, (a_{1},a_{2},a_{3}) \to a_{3}$

Answer 3

I find it illustrative to think in terms of subspaces. If your space has dimension $n$, just pick any $n - 1$ linearly independent vectors. Their span is a subspace of codimension $1$ (dimenions $n - 1$).

The hyperplanes, then, are just translations of these proper maximal subspaces; their cosets.

In your specific example, here are two rough drawings of a maximal proper subspace and one of its translates, both hyperplanes:

But don't be fooled; the $8$ points are all there is in $\Bbb Z_2^3$; the lines are just there to help us see the points as $\{0,1\}^n$.