$\therefore$">Above is point-line incidence geometry for the Fano Plane.
In other words, this matroid $M$ has a ground set $E(M)=\{ a, b, c, d, e, f, g\}$ and a collection of bases $\mathcal{B}(M)=\{X\subseteq E|X\neq \{a, e, b\}, X\neq\{b, g, c\}, X\neq \{c, f, a\}, X\neq \{e, d, c\}, X\neq \{f, d, b\}, X\neq \{a, d, g\}, X\neq \{e, g, f\}, \text{ and } |X|=3\}$.
$\textbf{Question:}$ This matroid $M$ is not representable as a vector matroid over the real numbers. I want to prove that it is not representable over the real numbers, but I am not sure how to go about doing this. Any help would be appreciated!
This is what I was have/was thinking so far for a proof...
Let $A$ be a matrix with column labels $a, b, c, d, f, g, e$ (i.e. 7 total).
As the rank of matrix is $3$ and $\{a, b, c\} \in \mathcal{B}(M)$, the first three columns can be simplified to this: $A=\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \end{bmatrix}$
As $\{c, d, f\}\in \mathcal{B}(M)$, I don't know where to go from here.