Let $M_i=(E,\mathcal I_i)$ be matroids with rank functions $r_i$ for $1\le i\le k$. The union $M_1\vee\cdots\vee M_k$ is a matroid, and its rank function is
\begin{align*}
r_{M_1\vee\cdots\vee M_k}(X)=\min_{A\subset X} \left(|X\setminus A|+\sum_{i=1}^k r_i(A)\right)
\end{align*}
for every $X\subset E$.