Practice Test: Test III,    Actual test is Wednesday, March 30, 2005

Monday, March 28, 2005
Math 307
Spring, 2005
Hentzel


1.   For the Markov Chain, find the steady state.  

         ______       .3       ______
         |     | ------------> |    |
         | A   | <---.2------  |  B |
         |_____| __       __   |____|
             \  /\         /| /
             .4   .3     .1  .5
               \   \     /  /
                \   \   /  /
                _\|  \ / |/_
                 __________
                 |   C    |
                 |________|


If at the start (A,B,C) = {100,100,100} what happens after 3 instances

of time?


2.  Find the change of basis matrix P   for the bases
                                     BC 

    | 1   2    0 |               |1    0    1  |
B = | 0   1    1 |          C =  |0    1    0  |
    | 0   1    0 |               |1    1    0  |





                       |  3    2    0  |
3.   Suppose   T    =  |  0    2    2  |
                BB     |  1    1    1  |

                     |  1  0  1  |
where the basis B is |  0  1  1  |
                     |  1  0  0  |.

                                         |  1  0  2  |
Find the matrix T     for the basis C is |  0  0  1  |.
                 CC                      |  1  1  0  |







4.   Solve the following differential equation.

          /
       |x|        | 15    12    0  -12  | | x |
       |y|    =   |  4     0   -2    0  | | y |
       |z|        | 24    25    3  -24  | | z |
       |w|        | 20    13   -2  -13  | | w |

     | 1   0   0   3 |
     | 0   1   2   0 |  
P =  | 2  -1  -1   6 |
     | 1   1   2   4 |

     | 4   3   0  -3 |
 -1  | 4  -1  -2   0 |
P  = |-2   1   1   0 |
     |-1  -1   0   1 |


5. Tidbits:

(a)  Fill in the bottom row to make this matrix stochastic.

        | .1  .2  .3 |
        | .4  .5  .7 |
        |  ?   ?   ? |  


(b)  If B and C are basis and T   is a linear transformation,
                               BB  

     Express T   in terms of B, C and T   .
              CC                       BB



(c)  What is the dimension of

           /   | 1 |    | 2 |   | 3 |  | 4 |  \
          /    | 5 |,   | 6 |,  | 7 |, | 8 |   \
          \    | 9 |    |10 |   |11 |  |12 |   /
           \   |13 |    |14 |   |15 |  |16 |  /



(d   What is the process to find the steady state of

     a stochastic matrix B?   


                        
(e)  Is the product of two stochastic matrices also stochastic? 



(f)  Prove that if a set V1, V2, ..., Vn  of vectors are dependent

     and V1 =/= 0, then some vector is a linear combination of the

     previous.


(g)  When is a matrix upper triangular matrix? 


(h)  Give 5 criterion that imply that an nxn matrix has an inverse.