Math 307
Spring, 2005
Hentzel

Time: 10:00-10:50 MWF
Room: 205 Carver 
Instructor: Irvin Roy Hentzel
Office 432 Carver
Phone 515-294-8141
E-mail: hentzel@iastate.edu

http://orion.math.iastate.edu/hentzel/class.307.05

Text:  Linear Algebra and its Applications, Third Edition
       David C. Lay




Practice Test: Test I,
Actual test is Monday, January 31, 2005
Hentzel

1.  Solve AX = B and write the answer as X = X  + a X  + a X  + ... a X  
                                              0    1 1    2 2        r r

and check your answer using A[X  X  X  ... X ] = [B 0 0 0 ... 0].
                               0  1  2      r

              |  1  2  0  1  2  3  | | x |     | 2 |
              |  1  3  0  0  1  2  | | y |     | 1 |
              |  2  5  0  1  3  5  | | z |   = | 3 |
              |  4 10  0  2  6 11  | | w |     | 7 |
                                     | u |
                                     | v |


2.  Find the inverse of this matrix.

            |  1  1  2  0  |
            |  1  2  3  0  |
            |  2  0  5  1  |
            |  2  3  4  0  |

3. (a)  Write the matrix of the linear transformation of differentiation

                           2x     2x   2 2x    3 2x
with respect to the basis e   , xe  , x e  ,  x e   .


(b)  Find some way to use the matrix from part (a) to compute the eighth

                          2    3   2x
derivative of {2 + 3x + 5x  - x } e   .


4.  Multiply these two matrices.

      | 1 0 0 0 1 0 | | 3 8 2 4 |
      | 0 1 0 0 0-1 | | 1 0 1 2 |
      | 0 0 2 0 0 0 | | 3 3 1 2 |
      | 1 1 1 0 0 0 | | 4 3 1 0 |
      | 0 0 0 0 1 1 | | 5 4 3 2 |
                      | 1 3 9 2 |



5. Tidbits:

(a)  The rows of AB are linear combinations of what? 

 
(b)  What is the rank of a matrix? 


(c)  What is the relation between the rank, the nullity, and the
 
     number of columns of a matrix?


(d)  What is the relationship between the Row Canonical Form of

     a matrix and the existence of the inverse of the matrix?


(e)  What is the 2x2 matrix which rotates the plane through 60 degrees?


(f)  Give a non zero 2x2 matrix which is not invertible. 


(g)  What are the three elementary row operations. 


(h)  When is a function a linear transformation?


(i)  Give two 2x2 matrices A and B such that AB =/= BA.


(j)  Give three 2x2 matrices A, B, C such that (AB)C =/= A(BC)