Let U_i denote the set of contrast vectors belonging to the main effect of factor i, U_{ij} the set of those belonging to the interaction between factors i and j, and more generally U_I the set of contrast vectors belonging to the interaction between the factors listed in I for any subset I \subset \{1, \ldots, k\} with |I| \ge 2 (here again |I| denotes cardinality). In addition, let U_{\emptyset} denote the set of constant vectors on T, that is, vectors whose components are equal. This defines a set U_I corresponding to each I \subset \{1, \ldots, k\}. It is not hard to see that each U_I is a vector space, a subspace of \R^n, where (as before) n=|T|, the number of treatment combinations. The following are well-known, fundamental facts:
