Some of the answers are included. 
Mathematics 2213:                                Linear Algebra

Final Exam                   1999 August 13           Version 4


NO CALCULATORS

Open book:  Books and Notes Allowed

Time:  Three Hours

Attempt Seven of the Ten Questions
   (Check that you have 10 questions)
   (If more than 7 questions are attempted, the best 7 will be used)
QUESTION 1
For this system of equations --

       1W +      5Y + 4Z = 0
            1X + 1Y + 6Z = 0
       1W +      5Y + 4Z = 0
            1X + 1Y + 6Z = 0

   a) Find two linearly independent solutions

                          W = _______  X= _______  Y= ________ Z = _______

                          W = _______  X= _______  Y= ________ Z = _______
   b) Find two more solutions.

                          W = _______  X= _______  Y= ________ Z = _______

                          W = _______  X= _______  Y= ________ Z = _______

   c) How many linearly independent solutions are possible.  Explain.



   d) Find two solutions for this system.

       1W +      5Y + 4Z = 4
            1X + 1Y + 6Z = 9
       1W +      5Y + 4Z = 4
            1X + 1Y + 6Z = 9

                          W = _______  X= _______  Y= ________ Z = _______

                          W = _______  X= _______  Y= ________ Z = _______
QUESTION 2

Starting with this statement,

"Every cow has 2 horns and 4 legs",
and using C as the number of cows, H the number of horns and, L the number of legs, express the statement as one or more equations. 
Comment:   Check the answer!   For example: 
Do the equations work when C=1, H=2, L=4 ? 

QUESTION 3

Evaluate these determinants

     |  1  0  0  4 |
     |  1  1  0  9 |  = ____________
     |  0  1  1 11 |
     |  0  0  1 10 |

     | 1 0 a |
     | 1 1 b |     =   _____________
     | 0 1 c |
Solutions:    4   and   (c-b+a) 
Comment: Using row operations is easier and less prone to error than expanding in minors. 

QUESTION 4

  Let U = [ 1 1 1 1 ]T  and  V = [ 4 9 1 7 ].
  The inverse of  I + U V   is  I + µ U V
  Evaluate  µ  as a number.
(I+µUV)(I+UV) 
=   I + UV + µUV + (µUV)UV 
=   I + UV + µUV + µU(VU)V 
=   I + UV + µUV + µU(21)V 
=   I + UV + µUV + (21)µUV 
=   I + (1 + 22µ)UV 
1+22µ = 0   when   µ = -1/22 

Comment:  In this question, take care not 
to commute matrix multiplications. 

QUESTION 5

Given n by n matrices A and B, n by 1 column vectors u and v, and that Au = 2v, and Bv = 3u.
  (a) Find an eigenvector of AB and determine its eigenvalue.
  (b) Find an eigenvector of BA and determine its eigenvalue.

Part (a) 
  (AB)v = A(Bv) = A(3u) = 3(Au) = 3(2v) = 6v 
  v is an eigenvector; its eigenvalue is 6

QUESTION 6 

a) Determine which of these are vector spaces. 
b) Determine the dimension of those which are vector spaces. 

The set of all --
 1) (w, x, y, z) for which   w=2x  and  y=2z
 2) (x, y)       for which   x² + y²=1
 3) (x, y)       for which   x + y = 0
 4) (x, y)       for which   x + y = 1
 5) (x, y, z)    for which   x=2y+1  and  y=2z+1
 6) (w, x, y, z) for which   w+x = y+z
 7) (x, y, z)    for which   x=2y  and  y=2z
 8) (x, y, z)    for which   x + 2y + 4z = 0
 9) (w, x, y, z) for which   w*x = y*z
 
*
1) Space of dimension 2
2) Not a vector space
3) Space of dimension 1
4) Not a vector space
5) Not a vector space
6) Space of dimension 3
7) Space of dimension 1
8) Space of dimension 2
9) Not a vector space
 
QUESTION 7
Express this matrix
as a product of
elementary matrices
 / 14 0 0 \
|   4 2 8  |
 \  0 0 1 /
(The answer is not unique.)

   .--------. .--------. .--------. .--------. .--------. .--------.
   |  |  |  | |  |  |  | |  |  |  | |  |  |  | |  |  |  | |  |  |  |
   |--+--+--| |--+--+--| |--+--+--| |--+--+--| |--+--+--| |--+--+--|
A= |  |  |  |*|  |  |  |*|  |  |  |*|  |  |  |*|  |  |  |*|  |  |  |
   |--+--+--| |--+--+--| |--+--+--| |--+--+--| |--+--+--| |--+--+--|
   |  |  |  | |  |  |  | |  |  |  | |  |  |  | |  |  |  | |  |  |  |
   `--------' `--------' `--------' `--------' `--------' `--------'
   The above should be elementary matrices.
Recommendation: 
Check the answer by multiplying the sequence of elementary matrices.

QUESTION 8

Let M be an n by n matrix, m a column vector, µ a scalar. Complete the cube of the partitioned matrix, A, in the space provided.

          | M : m |
    A =   |...:...|
          | 0 : µ |

          .-------------------------------------------.
          |                   :                       |
          |        3          :                       |
      3   |       M           :                       |
     A  = |...................:.......................|
          |                   :                       |
          |                   :                       |
          |                   :                       |
          `-------------------------------------------'
M3  (M2 + µM +µ3) m 
µ3 

QUESTION 9

A Least Squares Problem:
Consider an inconsistent system,  A*X = B, with this A and B:
A = 10     7     9            B = 142
    10     6     3                116
    10     5    -1                 98
    10     3    -3                106
    10     2    -1                100
    10     1     3                 94
 There are three candidates for least squares solution of A*X=B,
 column vectors called,  "Xa", "Xb", and "Xc".  One of them is the least
 squares solution.
    Xa = 9     Xb = 10      Xc = 8
         4           1           7
         2           3           1
 Calculate the squared norms ||A*X - B||² for each of the three
 candidate solutions. Use the norms to decide which of them is the
 least squares solution.

 To save arithmetic, A*X is calculated for the candidate X`s.
   A*Xa = 136       A*Xb  = 134         A*Xc = 138
          120               115                125
          108               102                114
           96                94                 98
           96                99                 93
          100               110                 90
  The best choice is Xa 
(a)    A Xa -   B =  residual
        136   142    -6
        120   116     4
        108    98    10
         96   106   -10
         96   100    -4
        100    94     6
                    304   = ||Ax.-B||2

(b)    A Xb -   B =  residual
        134   142    -8
        115   116    -1
        102    98     4
         94   106   -12
         99   100    -1
        110    94    16
                    482   = ||Ax.-B||2

(c)    A Xc -   B =  residual
        138   142    -4
        125   116     9
        114    98    16
         98   106    -8
         93   100    -7
         90    94    -4
                    482   = ||Ax.-B||2
 
QUESTION 10 
Suppose A geographic data base stores locations as latitude and
longitude in degrees. At Fredericton, a degree of latitude amounts to
50 miles and a degree of longitude to 69 miles. It is desired to make
a map of Fredericton for which
 a) The map is centered at 45.8 degrees North, 39.8 degrees West
    (roughly at the parliment buildings in Fredericton)
 b) Distances are in miles rather than degrees.
 c) The Y axis is parallel to Regent Street which runs in a direction
    30 degrees east of north.

Obtain a conversion formula to change a data base coordinate vector,
 u = | Latitude  | in degrees, to v = | X |, a coordinate vector
     | Longitude |                    | Y |
in miles from the center of the map:
     v = R*S*(u-b)      
where R a matrix to rotate, S is a matrix to scale,  and b an offset
vector to move the center.


Notes:
  *  R, S, and B should be in mumeric form.
  *  Latitude is distance North of the Equator in degrees.
  *  Longitude is distance West of Greenwich England in degrees.
  *  sin(30 deg)=0.5     cos(30 deg)=0.8660

      .-----------.            .-------------.           .-----.
      |     |     |            |      |      |           |     |
      |     |     |            |      |      |           |     |
  R = |-----|-----|        S = |------|------|       b = |-----|
      |     |     |            |      |      |           |     |
      |     |     |            |      |      |           |     |
      `-----------'            `-------------'           `-----'
   R =   cos 30  -sin 30    =   0.8660  -0.5
         sin 30   cos 30        0.5      0.8660

   S =   50   0
          0  69

   b =   45.8
         39.8