Summary: Matroid theory gives us powerful techniques for understanding combinatorial optimization problems and for designing polynomial-time algorithms. However, several natural matroid problems, such as 3-matroid intersection, are NP-hard. Here we investigate these problems from the parameterized complexity point of view: instead of the trivial \(n^{O(k)}\) time brute force algorithm for finding a \(k\)-element solution, we try to give algorithms with uniformly polynomial (i.e., \(f(k)\cdot n^{O(1)})\) running time. The main result is that if the ground set of a represented linear matroid is partitioned into blocks of size \(\ell \), then we can determine in randomized time \(f(k,\ell )\cdot n^{O(1)}\) whether there is an independent set that is the union of \(k\) blocks. As a consequence, algorithms with similar running time are obtained for other problems such as finding a \(k\)-element set in the intersection of \(\ell \) matroids, or finding \(k\) terminals in a network such that each of them can be connected simultaneously to the source by \(\ell \) disjoint paths.