In Jürgen Elstrodt's Measure and Integration Theory (8th German edition) there is the following exercise I.3.6:
Exercise: The group $(\mathfrak{P}(X), \Delta)$ (power set with symmetric difference) is in a natural way a vector space over the field $\mathbb{Z}/2\mathbb{Z}$. Let $X$ be finite, $V$ a sub-vector space of this vector space and for $A, B \in V$ let $\langle A, B\rangle := \lvert A \cap B\rvert + 2\mathbb{Z}$, where $\lvert A \cap B\rvert$ denotes the number of elements of $A \cap B$. Show that $\langle{\cdot},{\cdot}\rangle \colon V \times V \to \mathbb{Z}/2\mathbb{Z}$ is a symmetric bilinear form. When is this bilinear form non-degenerate?
Showing that $\langle{\cdot},{\cdot}\rangle$ is a symmetric bilinear form is not hard. But I don't find a good criterion for the non-degeneracy on $V$, apart from the obvious reformulation “for every $A \in V$ with $A \neq \emptyset$ there exists a $B \in V$ such that $A \cap B$ contains an odd number of elements”. Is there a better criterion?
An enumeration $X \xrightarrow{\sim} \{1, \dots, n\}$ leads to an identification of $\mathfrak{P}(X)$ with the vector space $(\mathbb{Z}/2\mathbb{Z})^n$ and under this identification the bilinear form of the exercise corresponds to the “standard scalar product” $(x, y) \mapsto \sum_{i = 1}^n x_i y_i$. But I can't see how this of any help. For example the restriction of the standard scalar product from $(\mathbb{Z}/2\mathbb{Z})^3$ to the sub-space $$\left\{\begin{pmatrix}0\\0\\0\end{pmatrix}, \begin{pmatrix}0\\1\\1\end{pmatrix}, \begin{pmatrix}1\\0\\1\end{pmatrix}, \begin{pmatrix}1\\1\\0\end{pmatrix} \right\} \subset (\mathbb{Z}/2\mathbb{Z})^3$$ is non-degenerate, but the restriction to $$\left\{\begin{pmatrix}0\\0\\0\end{pmatrix}, \begin{pmatrix}0\\1\\1\end{pmatrix}, \begin{pmatrix}1\\0\\0\end{pmatrix}, \begin{pmatrix}1\\1\\1\end{pmatrix} \right\} \subset (\mathbb{Z}/2\mathbb{Z})^3$$ is degenerate. So the dimension of $V$ alone is certainly not enough to read of the non-degeneracy. I am hoping for hints or comments!