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Question 4.E.2.12: For matrices such that the product AB is defined, explain wh......

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).

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(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).

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