Spaces of matrices without non-zero eigenvalues in their field of definition, and a question of Szechtman

Let V be a vector space of dimension n over any field F. Extreme values for the possible dimension of a linear subspace of EndF(V) with a particular property are considered in two specific cases. It is shown that if E1 is a subspace of EndF(V) and there exists an endomorphism g of V, not in E1, such that for every hyperplane H of V some element of E1 agrees with g on H, then E1 has dimension at least . This answers a question that was posed by Szechtman in 2003. It is also shown that a linear subspace of Mn(F) in which no element possesses a non–zero eigenvalue in F may have dimension at most . The connection between these two properties, which arises from duality considerations, is discussed.