Let A and B be two matrices whose product is defined. (a) Show that row(AB)⊂row(B). (Hint, start with showing that rows of AB are linear combination of rows of B). (b) Use part (1) and transpose to show that col(AB)⊂col(A). (c) Find an example to show that row space of AB needs not to be equal to row space of B. (d) Show that if A invertible then the equality holds, i.e. row(AB)=row(B) (Hint: Use (1) and B=A −1
(AB)) (e) Use (4) to give another proof for Theorem 3.20. Let B be any matrix that is row equivalent to a matrix A. Then row (B)=row(A).