'Dot Product' on a Vectors Space over $\mathbb{F}_2$ Let $\mathbb{F}_2$ be the field with 2 elements, and consider $\mathbb{F}_2^n$, the space of all $n$-tuples over $\mathbb{F}_2$. This is an $n$-dimensional vector space over $\mathbb{F}_2$, and we may introduce the following 'dot product' for every pair of vectors $x,y$ in $\mathbb{F}_2^n$ ; if $x=(x_1,\ldots,x_n)$ and $y=(y_1,\ldots,y_n)$, we define
$$\langle x,y\rangle:=x_1y_1+\cdots+x_ny_n=\sum_{i=1}^{n}x_iy_i$$
Clearly this is a symmetric bilinear form, and my question is : when does this form become non-degenerate? More precisely,

For which subspaces $V$ (of $\mathbb{F}_2^n$) does the induced bilinear form $\langle \cdot,\cdot\rangle:V\times V\rightarrow\mathbb{F}_2$ become non-degenerate?

I've been struggling with this problem for quite a while, but still don't see how to approach in the right way. Any advice is welcome.
 A: It will be easier to talk about subspaces on which it becomes degenerate. This means subspaces $V$ containing a nonzero vector $v$ such that $\langle v, w \rangle = 0$ for all $w \in V$. In particular, $v$ must satisfy $\langle v, v \rangle = 0$; such vectors are called null vectors or isotropic vectors, and for this particular bilinear form they are precisely the vectors $v \in \mathbb{F}_2^n$ with an even number of nonzero entries. 
Given a nonzero vector $v$, the orthogonal complement 
$$v^{\perp} = \{ w \in \mathbb{F}_2^n : \langle v, w \rangle = 0 \}$$ 
can be explicitly described as follows. If $v$ has nonzero entries in the indices $i_1, \dots i_k$, then $\langle v, w \rangle = 0$ iff among the $i_1, \dots i_k$th entries of $w$ an even number of them are nonzero. 
Every degenerate subspace $V$ can be constructed by taking an isotropic vector $v$ and then letting $V$ be any subspace of $v^{\perp}$ containing $v$. This isn't quite as good as a classification, though. It would be nice to be able to count the degenerate subspaces, since there are finitely many of them. One relevant keyword here should be "isotropic Grassmannian." 
