Possible dimensions of subspace intersections for five direct summands

The paper consides the problem on the dimensions of the intersections of a subspace in the direct sum of a finite series of finite-dimensional vector spaces with the sums of pairs of direct summands, provided that the subspace intersection with each of these direct summands is empty. The problem is naturally divided into two ones: Find conditions for the existence and for the representability of the corresponding matroid. The following theorem is proved: If the ranks of all the unions of a series of blocks satisfying the condition for the ranks of subsets in the matroid are given and the blocks have full rank, then this partial rank function can be extended to a full rank function for all the subsets of the base set (the union of all the blocks). Necessary and sufficient conditions on the dimensions of the direct summands and intersections mentioned above for the corresponding matroid to exist are obtained in the case of five direct summands.