Prove (where A is an nn matrix and so defines a transformation of any n-dimensional space V with respect to B, B where B is a basis) that dim(R(A) ∩ N (A)) = dim(R(A)) - dim(R(A2)). Conclude
(a) N (A) ⊂ R(A) iff dim(N (A)) = dim(R(A)) - dim(R(A2)),
(b) R(A) ⊆ N (A) iff A2 = 0,
(c) R(A) = N (A) iff A2 = 0 and dim(N (A)) = dim(R(A)),
(d) dim(R(A) ∩ N (A)) = 0 iff dim(R(A)) = dim(R(A2)),
(e) (Requires the Direct Sum subsection, which is optional.) V = R(A) ⊕ N (A) iff dim(R(A)) = dim(R(A2)).