Here are the answers for last semester's midterm. Full solutions
are availible in the class binder on reserve in Cabot.

1.  a) If k = 1 then no solution.
    b) If k is not equal to 0 or 1 then a unique solution.
    c) If k = 0, then there are 1-parameters worth of solutions: 
         (0, t, -1)

    d), e) do not happen.

2.  a) Basis for nullspace = { (-t, 0, t) }, nullity = 1
    b) rank = 2, collumn space is all of R^2, so one choice
       of basis is {(1,0), (0,1)}.

3.  Let A = [ 1 0 0 ],  y = [ 2 ], then [ a ]
            [ 1 1 1 ]       [ 2 ]       [ b ] = (A^T A)^(-1) A^T y
            [ 1 2 4 ]       [ 1 ]       [ c ] 
            [ 1 3 9 ]       [ 3 ]

4.  a) (2, 3)  b)  [ 0  -1 ]   c)  [ 0  1 ]
                   [ 1   0 ]       [-1  0 ]

5.  a) For P1: x + y + z = 3, For P2: x + y + z = 5
    b) Parallel because normal vectors are scalar 
       mults of each other.
    c) No, because the vector joining p to q does not meet P_1 in 
       in right angles.

6. a) (0,0) saddle, (-1,-1) local max

7.  Crit pts: (0,0), (1,0), (-1,0), (1, -2), (-1, 2)
    Global maxes: (1,0), (-1,0)
    Global mins: (1, -2), (-1, 2)

8.  F T F T F F F F 

9.  a) eigenvalues: 1, 2; cor. eigenvectors (1, 1), (2, 1)
    b) 
       [  2^17 - 1  -2^17 + 2 ]
       [                      ]
       [  2^16 - 1  -2^16 + 2 ]

10.  a)  [ 2/3  1/6 ]   b) Pop. stabilizes at (4000, 8000)
         [          ]
         [ 1/3  5/6 ]

     c)  [ 1/3  1/4 ]   d) Pop increases exponentially at 29% a year.
         [          ]
         [ 1/6  5/4 ]

11. (1, 1) is an eigenvector with e-value 2.
    (-1, 1) is an eigenvector with e-value 1/2.
    Determinant is 1.