§16 — the seven-element matroid $M'$ corresponds to no real matrix

Let $M'$ be Whitney's seven-element matroid of §16: its elements are $1,\dots,7$ and its bases are all sets of three elements except

Then:

1. such a matroid exists (the postulates for rank are satisfied);
2. no real matrix corresponds to it: for every $m\ge 0$ and every real $m\times 7$ matrix $\mathbf M$, the matroid of $\mathbf M$ (columns as elements, rank of a column set = rank of the submatrix) is not $M'$.

$M'$ is the Fano matroid, the matroid of the projective plane over the field with two elements. The theorem gives the first example of a matroid that is not representable over the reals, showing that Whitney's abstract postulates are strictly more general than linear dependence of real vectors.

Formalization Note The elements are Fin 7 (Whitney's $k$ is k - 1). The matrix has an arbitrary number $m$ of rows and its columns are labelled by the seven elements; since all matrices are quantified over, a relabelling of columns is already covered. The existence clause rules out a vacuous non-existence statement.