Question

Let F be a field, and let V = F(N, F). Define T :V → V by T(f) :N → F for f ∈ V
as given by T(f)(i) = f(i+1) for each i ∈ N. Prove that T is a linear transformation (done),
and that R(T) = V (so N(T) ⊆ R(T) and V = R(T) = R(T) + N(T)), and find a
basis for N(T) (showing in particular that N(T) 6= { 0 }).