On spaces of matrices containing a nonzero matrix of bounded rank

Let Mn(R) and Sn(R) be the spaces of n × n real matri-ces and real symmetric matrices respectively. We continue to study d(n, n − 2,R): The minimal number ` such that every `-dimensional subspace of Sn(R) contains a nonzero matrix of rank n−2 or less. We show that d(4, 2,R) = 5 and obtain some upper bounds and monotonicity properties of d(n, n − 2,R). We give upper bounds for the dimensions of n − 1 subspaces (subspaces where every nonzero matrix has rank n − 1) of Mn(R) and Sn(R), which are sharp in many cases. We study the subspaces of Mn(R) and Sn(R) where each nonzero ma-trix has rank n or n − 1. For a fixed integer q> 1 we find an infinite sequence of n such that any (q+1 2 dimensional sub