Decompositions of linear spaces induced by bilinear maps

Let V be an arbitrary linear space and f : V x V -> V a bilinear map. We show that, for any choice of basis B of V, the bilinear map f induces on V a decomposition V = circle plus V-j is an element of J(j) as a direct sum of linear subspaces, which is f-orthogonal in the sense f (V-j, V-k) = 0 when j not equal k, and in such a way that any V-j is strongly f-invariant in the sense f (V-j, V) + f (V, V-j) subset of V-j. We also characterize the f-simplicity of any V-j. Finally, an application to the structure theory of arbitrary algebras is also provided. (C) 2017 Elsevier Inc. All rights reserved.