HW #3, due 3:50pm Friday, Feb 23

1.9.38 (Is transformation 1-to-1?

1.9.40 (Is transformation onto.)

1.10.12 (Rental car migration.)

Note: this kind of a model really only applies well to systems where the numbers of items are large, so that effects of randomness are not important. A bit dubious with the numbers in this exercise.

2.1.12 (Demonstration that AB=0 does not imply A=0 or B=0.)

2.1.22 (L.D. of columns of B implies L.D. of columns of AB.)

Hints: The approach here is to "unpack" the two statements "columns of B are L.D." and "columns of AB are L.D." (I mean translate each of them to a sentence that begins "There exists ..") and then show the former implies the latter. To do this, you will need to invoke the associativity of matrix multiplication.

2.2.X (Inverse of product.) This was accidentally not published until quite late, so will be ncluded in Homework #4 instead. Your random matrix should of course be different from mine, and that of all other students.

a. Construct 2 largish (at least 5x5) random square matrices, A and B, with integer values. Form the product AB.

b. Use the algorithm I showed you in class on Feb 20 to find the inverses of A, B and AB.

c. Show that, at least in this case, the inverse of AB is the product of the inverses of A and B in reverse order.