1. Let V be a vector space and F: V R a linear map. Let W be the subset of V consisting of all elements v such that F(v)=0. Assume that W V, and let be an element of V which does not lie in W. Show that every element of V can be written as a sum w + c , with some w in W and some number c.
2. In exercise (1), show that W is a subspace of V. Let { } be a basis of W. Show that { } is a basis of V.