  A matroid $M$ is an ordered pair $(E,I)$, where $E$ is a finite set called
the ground set and a collection $I\subset 2^{E}$ called the independent sets
which satisfy the conditions: (i) $\emptyset \in I$, (ii) $I'\subset I \in I$
implies $I'\in I$, and (iii) $I_1,I_2 \in I$ and $|I_1| < |I_2|$ implies that
there is an $e\in I_2$ such that $I_1\cup \{e\} \in I$. The rank $rank(M)$ of a
matroid $M$ is the maximum size of an independent set. We say that a matroid
$M=(E,I)$ is representable over the reals if there is a map $\varphi \colon E
\rightarrow \mathbb{R}^{rank(M)}$ such that $I\in I$ if and only if
$\varphi(I)$ forms a linearly independent set.
  We study the problem of matroid realizability over the reals. Given a matroid
$M$, we ask whether there is a set of points in the Euclidean space
representing $M$. We show that matroid realizability is $\exists \mathbb
R$-complete, already for matroids of rank 3. The complexity class $\exists
\mathbb R$ can be defined as the family of algorithmic problems that is
polynomial-time is equivalent to determining if a multivariate polynomial with
integers coefficients has a real root.
  Our methods are similar to previous methods from the literature. Yet, the
result itself was never pointed out and there is no proof readily available in
the language of computer science.
