For each question below, indicate whether the statement is true or false. If it is true, explain why. If it is false, provide a counter-example.

Question

a. (2 points) True or False: If x is a vector in the null space of A then x will also be in the null space of M = AB. b. (2 points) True or False: If x is a vector in the null space of B then x will also be in the null space of M = AB. c. (2 points) True or False: If x is a vector in the null space of M = AB then x will also be in the null space of A. d. (2 points) True or False: If x is a vector in the null space of M = AB then x will also be in the null space of B.

Answer 1

a. False. Counterexample:

Let A = {{1, 1}, {0, 0}}, B = {{-1, 1}, {0, 0}}, and x = {{-1}, {1}}.

Then, x is in the null space of A, since Ax = 0.

However, Bx = {{2}, {0}}, so Mx = ABx = {{1, 1}, {0, 0}} * {{2}, {0}} = {{2}, {0}} != 0

Therefore x is not in the null space of M = AB.

b. True. Proof:

x is in the null space of B, so Bx = 0

So, (AB)x = A(Bx) = A*0 = 0.

c. False. Counterexample:

Let A = {{1, 1}, {0, 0}}, B = {{-1, 1}, {0, 0}}, and x = {{1}, {1}}.

Then, x is in the null space of AB, but only because x is in the null space of B, as Bx = 0, meaning ABx = A*0 = 0.

x is not in the null space of A, since Ax = {{2}, {0}}.

d. False. Counterexample:

Let A = {{-1, 1}, {0, 0}}, B = {{1, 0}, {1, 0}}, and x = {{1, 0}.

Then, ABx = 0, but Bx = {{1}, {1}} != 0.