2. (10 points total) Pascal and Serge are discussing ranges and null spaces. (a) (3 points) Pascal says the following: Let's say we have two matrices A and B such that BA is defined. Then null (BA) ( null(B), and here is the proof: The null space of B consists of those vectors that B sends to O. Now if B sends a vector to zero already, then of course BA will also send it to O. Point out all the mistakes Pascal made in 3 sentences or less, and give a concrete counterexample. (b) (2 points) After refuting Pascal's argument, Serge thinks that there is a similar looking statement about the null spaces of BA and B where a similar argument would work, but he can't quite come up with the right one. State a correct subset relation of the form: "BLANK C BLANK", where either of the blanks are the null spaces of B, A, and BA (you are not allowed to use the same null space twice), then give an argument for your claim. (c) (2 points) Pascal then makes another claim: Again assume that A and B are matrices such that BA is defined. Then col( BA) C col( B). Here is my proof: The column space of BA consists of vectors that can be written as linear combinations of the columns of B and the columns of A. Therefore the columns of B are expressible as linear combinations of the columns of BA, so we have col ( BA] C col(B). Point out all the mistakes Pascal made in his argument in 3 sentences or less. (d) (3 points) Serge no longer trusts Pascal's arguments, and he is not sure if the claim col(BA) ( col(B) is correct. Is the statement col(BA) C col(B) true? If no, provide a concrete counterexample, if yes, give an explanation