SPACES OF SYMMETRICAL MATRICES OF BOUNDED RANK

Let I(n)(F) denote the space of all n X n symmetric matrices over a fi eld F. Let t be a positive integer such that t < n. A subspace W of I( n)(F) is said to be a tBAR-subspace provided that the rank of every ma trix in W is bounded by t. Meshulam showed, under the assumption \F\ g reater-than-or-equal-to n + 1, that the maximal dimension of a tBAR-su bspace of I(n)(F) is given by max{(t + 1/2), (k + 1/2) + k(n - k)} if t = 2k, max{(t + 1/2), (k + 1/2) + k(n - k) + 1} if t = 2k + 1. Provid ed that we also assume char F not-equal 2, we show here that any tBAR- subspace of I(n)(F) of maximal dimension is congruent to W1(n, t) = {A is-an-element-of I(n)(F):a(ij) = 0 if i > t or J > t}, or [GRAPHICS] Which of the two possibilities occurs depends on the values of n and t .