Part 14a: SVD returns
These are extra problems, some of which are taken from section 7.3
which uses a lot of SVD, though that section won't be explicitly 
on the final.

1: Find the singular value decomposition of the invertible matrix:
   A= [3, 3; -1, 1]

2: Rewrite the above SVD as a sum 
   A = sigma_1 u_1 v_1^T + sigma_2 u_2 v_2^T.
   Check that this equation is true.

3: Show from rewriting the SVD relation A=U*(Sigma)*V^T that every
   matrix of rank r is the sum of r matrices of rank 1 with form sigma_i u_i v_i^T.

4: Find the singular value decomposition of the non-invertible matrix:
   A= [3 4 0]
Part 14b: Applications

1: Suppose we play a board game with 10 spaces in a loop.  Each player rolls
   a single dice and moves that number of squares.  Ending a move on "Go to Jail" 
   (square number 4) moves you to "Jail" (square number 9).  Ending a move on 
   "Chance" (square 8) has you flip a coin: tails you stay, but heads move to "Jail" 
   (square 9).  

   Write the matrix for this Markov process where the state of the process 
   is which square you end your turn on.
   (Hint: divide the process into three "rules" each with its own matrix.
   A Markov matrix means no negative entries and columns sum to 1.)

2: Draw the network with connections listed below. 
   (Each node is assigned a number.  Connections go both directions).
   Write the adjacency matrix for this network (see wikipedia for details if you need to).
   It should have elements 1 and 0 with A_ij = 1 when node i and j are connected.

   Connections:     1-3, 1-5, 2-4, 2-3, 6-2, 3-6, 4-6, 4-5.
   Why is this matrix symmetric?
Study for the exam by 
  1) reading the text and lecture notes, 
  2) doing problems from the list of good problems, 
  3) making an outline of the course so far and 
  4) creating example problems of your own for the exam.