Let A be an m x k-matrix, and let B be a k x n-matrix. (a)...

 

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Let A be an m x k-matrix, and let B be a k x n-matrix. (a) Prove that the column space of AB is contained in the column space of A. Subset proof format: Prove the implication (vector v) v € Col(AB) ⇒ (vector v) v E Col(A). (b) Assume that k = m and that A is invertible. Prove that the null space of AB is equal to the null space of B. Proof format for equality of sets: prove (vector x) x € Null(AB) = (vector x) x € Null(B). (c) in the situation of question part (b), prove that the rank of AB is equal to the rank of B. Let A be an m x k-matrix, and let B be a k x n-matrix. (a) Prove that the column space of AB is contained in the column space of A. Subset proof format: Prove the implication (vector v) v € Col(AB) ⇒ (vector v) v E Col(A). (b) Assume that k = m and that A is invertible. Prove that the null space of AB is equal to the null space of B. Proof format for equality of sets: prove (vector x) x € Null(AB) = (vector x) x € Null(B). (c) in the situation of question part (b), prove that the rank of AB is equal to the rank of B.