Let $E$ be a finite set, and let $\mathcal B \subseteq 2^E$ be a nonempty collection of subsets of $E$. Then $\mathcal B$ is the collection of bases of a matroid on $E$ if and only if the following two conditions hold:
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1. For all $B_1, B_2 \in \mathcal B$, if $B_1 \subseteq B_2$, then $B_1 = B_2$.
2. For all $B_1, B_2 \in \mathcal B$ and every $e \in B_1 \setminus B_2$, there exists $f \in B_2 \setminus B_1$ such that
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\begin{align*}
(B_1 \setminus \{e\}) \cup \{f\} \in \mathcal B.
\end{align*}
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When these conditions hold, the independent sets of the corresponding matroid are exactly
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\begin{align*}
\mathcal I := \{I \subseteq E : I \subseteq B \text{ for some } B \in \mathcal B\}.
\end{align*}