1. Consider Hasse diagram
(a) List the pairs of elements that are not related via this partial order.
(b) Construct, as far as possible, the table of least upper bounds for pairs of elements under this partial order.
(c) Is this poset a lattice? (provide an argument for your answer).
(d) Let p b), (a, b), (b, a), (c, c), (c, d), (d, c)} be a relation on the set {a, b,c, d}, Construct p2, list explicitly all the ordered pairs that form p2.
(e) Draw a directed graph diagram for p and p2. Which ordered pair(s) must be adjoined to p2 to complete it into an txluivalence relation on the set {a, b, c, d}?
(2 + 5 + 3 4 + = 18 marks)
2. For the following system of linear
(a) Use Gaussian elimination to put the system into row echelon form
(b) Use Gauss-Jordan elimination to change your row echelon form into reduced row echelon form.
(c) Solve the system of equations using either your answer to part (a) or your answer to part
(d) Verify that your solutions satisfy this system of equations.
= IS marks)
3. Consider matrix
3
1
—2
(a) Confirm that the matrix A is invertible.
(b) Find all the cofactors Cij of the matrix A and hence find adj(A).
(c) Verify that the adj(A) you obtained is correct by multiplying it with A.
— 10 marks)
4. For the following matrix
(a) Find the eigenvalues (one eigenvalue is Al = 2)
(b) For each eigenvalue find the corresponding eigenvector(s).
(c) Determine if possible a matrix p so that B = p¯IAP is in diagonal form. Write down B and p.
(2 + S •F 6 = 16 marks)
5. The vector space of solutions of Ax = O
is generated by
(a) Verify that each vector is a solution.
(b) Show that any solution can be written as a linear combination
for a suitable choice Of q , 02, c].
(c) What is the dimension of the row-space of A and the dimension of its nullspace? State one basis for the nullspace of A.
(6 + 8 8 = 22 marks)
6. For matrix
(a) Determine the row-rank.
(b) Find a set of generators for the row space of A.
(c) State one possible basis for the row space of A and explain why it is a basis.
= 16 marks)
7. The 6-tuples:
= 011100, u2 = 111010, = 110011 form a biLSis for a (6,3) linear code.
(a) Write down the generator matrix for this code.
(b) Construct code words for the message blocks: 101, 110, 011, 010.
(c) Construct the parity check matrix for this code.
(d) Decode if possible
(6+ 4 + + 20 marks)
