For matrices such that the product AB is defined, explain why each of the following statements is true.
(a) R (AB) ⊆ R (A).
(b) N (AB) ⊇ N (B).
(a) b ∈ R(AB) \Longrightarrow ∃ x such that b = ABx = A(Bx) \Longrightarrow b ∈ R(A) because b is the image of Bx.
(b) x ∈ N (B) \Longrightarrow Bx = 0 \Longrightarrow ABx = 0 \Longrightarrow x ∈ N (AB).