The following results are proved. The Null Space TheoremLet X,Y be vector spaces, P∈L(X),Q∈L(Y) be projections and T∈L(X,Y) be invertible. (The restriction of QTP to R(P) can be viewed as a linear operator from R(P) to R(Q). This is called a section of T by P and Q and will be denoted by TP,Q.) Then there is a linear bijection between the null space of the section TP,Q of T and the null space of its complementary section TIY-Q,IX-P-1 of T-1. Let X be a Banach space with a Schauder basis A={a1,a2,⋯}. Let T be a bounded (continuous) linear operator on X. Suppose the matrix of T with respect to A is tridiagonal. If T is invertible, then every submatrix of the matrix of T-1 with respect to A that lies on or above the main diagonal (or on or below the main diagonal) is of rank ≤1. © 2015 Elsevier Inc. All rights reserved.