Practice Test: Test II,    
               Actual test is Wednesday, February 23, 2005 

Monday, February 21, 2005
Math 307
Spring, 2005
Hentzel

1.    Write the 3x3 matrix which rotates the space

60 degrees around the line  (1,2,2)












2.    A working economy has three divisions R,S,T.

The current data is


          R       S       T     Demand      Total
    R    30      20      10       20          80 
    S    10       0      20       30          60
    T    50      40       0       30         120

                        | r |
What should the Total = | s |  be if the Demand vector is
                        | t |

           | 121 |
changed to | 121 |.
           | 121 |












3.  Prove that the Row Rank of any matrix is the same as

the column rank. 















4.   For the matrix A given below, find a basis of the

range and a basis of the kernel.

          | 1  2  0  1  0 |
          | 0  1  1  0  1 |
      A = | 2  3  1  1  1 |
          | 0  0  0  0  1 |











5. Tidbits:

(a)  Write the 2x3 matrix for rotation through an angle of 45 degrees.




(b)  For a matrix A      with n > m.  Which has the bigger
                    nxm 

dimension, the row space of A or the column space of A?



                            | 0 1 0 | 
(c) How does the matrix E = | 1 0 0 | change the matrix 
                            | 0 0 1 |
A in the product   EA?


                            | 0 1 0 | 
(d) How does the matrix E = | 1 0 0 | change the matrix 
                            | 0 0 1 |
A in the product   AE?


                                            | x  y |
(e)   What is the inverse of the matrix     | z  w | ?



(f)   What is the null space of a matrix?


(g)   What is the rank of a matrix? 


(h)   When is a subset a vector space?