Name: Student ID:
Sample Final exam questions (mandatory part)
The actual final exam will have at most 9 mandatory questions and 5 optional questions. The optional questions will be
similar to the ones on the (sample) tests, and need to be answered only if you do not want me to re-use your average test
score. The list of questions below is supposed to help you prepare for the mandatory part of the final.
1. Write up the multiplication table of Z7.
2. Provide a formula for the number of bases in an n-dimensional vector space over GF[q]and prove its validity.
3. What is the number of 3-dimensional subspaces of a 7-dimensional space over the field Z5=GF [5]?
4. Identify each subset of a set Swith its characteristic vector over Z2. What set-operation corresponds to taking the
sum of two characteristic vectors? What vector operation corresponds to taking the intersection of two sets?
5. List the elements of the cycle subspace and of the cutset subspace for the graph shown in Fig. 5.6 (page 63) of our
textbook.
6. Assume that C1= [A1, B1]and C2= [A2, B2]are cuts. Prove that their symmetric difference is a disjoint union
of cuts. (Draw a picture.) Use this observation to describe the elements of the cutset subspace.
7. Describe the elements of the cycle subspace of a graph.
8. Prove that a graph is a disjoint union of cycles if and only if the degree of every vertex is even. How does this
observation help to prove that the cycle subspace is closed under vector addition?
9. In the proof of the description of the cycle space and of the cutspace we never showed closure under scalar multi-
plication. Why? How can you describe subspaces of a vector space over the field Z2?
10. Explain how a spanning tree may be used to find a basis of the cycle and cutset spaces. Illustrate the method by
listing the fundamental cycles and cuts for the graph shown below:
4
1
2
3
9
5
8 7
6
Figure 1: Graph with selected spanning tree
11. Based on the answer to the previous question, what can you say about the dimension of the cycle space and the
cutset space? Are these spaces complements of each other?
MATH 3116 GRA PH THE ORY Fall 2005
