Example of where the sum of a subspace and its orthogonal complement is not the original vector space?

Question

Suppose $\mathbb{F}$ is an arbitrary field and let $W$ be a subspace of $\mathbb{F}^n$. $W^\perp$ can be defined in exactly the same way as in the real case.

Show by example that it isn't necessarily true that $W+W^\perp = \mathbb{F}^n$.

Answer 1

Presumably, the intended bilinear form on $\mathbb{F}^n$ with respect to which $W^\perp$ is defined is the map $B:\mathbb{F}^n\times\mathbb{F}^n\to\mathbb{F}$ defined by $B(x,y)=\sum_{i=1}^nx_iy_i$, so that $$W^\perp=\{x\in\mathbb{F}^n:B(x,w)=0\text{ for all }w\in W\}$$ (See the relevant Wikipedia page.)

For a simple counterexample, let $\mathbb{F}$ be the field with two elements, and let $n=2$. I leave it to you to find the subspace $W\subset \mathbb{F}^n$ with the property that $W+W^\perp\neq\mathbb{F}^n$.