Prof. E. Berlekamp
                                September 21, 1999
                Math 110 QUIZ



        Summary:

        Problem 1: LU factorization             30 points
        Problem 2: Binary erasure correction    30 points (10 pts/part)
        Problem 3: Basis and Dimension          40 points (10 pts/part)
        Problem 4: Rudimentary operator algebra 20 points 


1) Without doing any permutations on the following real matrix,
factor it into LU, where L is lower-triangular, and U is
upper triangular.  L should have all ones on its main diagonal.


        2  1  0  0  0
        3  2  1  0  0
        0  3  2  1  0
        0  0  3  2  1
        0  0  0  3  2



2) The "Hamming" code is a binary vector space, defined
as the nullspace of the following 4x8 binary matrix:

            0 0 0 0 1 1 1 1
      H  =  0 0 1 1 0 0 1 1
            0 1 0 1 0 1 0 1
            1 0 0 1 0 1 1 0

The vectors in this code are called codewords.  Any codeword
is a possible message.  Transmission noise might cause some
bits of a message to be "erased", and replaced with "?"s.

2a)  Suppose the received word is

x0   0
x1   ?
x2   1
x3 = 1
x4   ?
x5   0
x6   ?
x7   1

If possible, using any method you like, find the values of
the bits in the three erased locations: x1, x4, and x6.


2b)  If possible, find a set of 4 locations which, if erased, can
be uniquely decoded.

2c)  If possible, find a set of 4 locations which, if erased, can
NOT be uniquely decoded.




3) Each part of this problem defines a vector space or subspace.
Determine its dimension, and exhibit a basis.

3a) 
The space of all vectors in R^4 whose components add to zero



3b)
The nullspace of the following real matrix:

     1 2 0 1 2 1
U =  0 0 2 2 0 0
     0 0 0 0 0 1
     0 0 0 0 0 0



3c) 
The space of all symmetric 3x3 matrices.



3d) 
The subspace of R^infinity which consists of all arithmetic
progressions, including

(1/2, 3/2, 5/2, 7/2, 9/2,... )

( 10,   7,   4,   1,  -2,... )

(1 7,  18,  19,  20,  21,... )




4) Let I denote the square nxn identity matrix, and let J
denote the nxn matrix which has "1" is each of its n^2
components:
       
       1 1 1 ... 1
       1 1 1 ... 1
  J =  1 1 1 ... 1
       ..... ... .
       1 1 1 ... 1


Let b be a scalar.  In terms of b and n, determine the value
of another scalar, c, such that (I + bJ) is the inverse of
(I + cJ).