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SIMON FRASER UNIVERSITY
DEPARTMENT OF MATHEMATICS
Midterm 2
MATH 232 Fall 2010
Instructor: Prof. JF Williams
Nov. 12 2010, 11:30 – 12:
Name: (please print)
family name given name(s)
SFU IDs:
student number @sfu.ca e-mail
Signature:
Instructions:
- Do not open this booklet until told to do so.
- Write your name above in block letters. Write your
SFU student number and email ID on the line pro-
vided for it.
- Write your answer in the space provided below the
question. If additional space is needed then use the
back of the previous page.
- This exam has 5 questions on 5 pages (not includ-
ing this cover page). Once the exam begins please
check to make sure your exam is complete.
- Calculators are not allowed. No books, papers, or
electronic devices of any kind shall be within the
reach of a student during the examination.
- During the examination, communicating with,
or deliberately exposing written papers to the
view of, other examinees is forbidden.
Question Points Score
Total 60
1. This question concerns a matrix A with characteristic polynomial
p(λ) = (λ − 1)
2 (λ + 4).
[4] (a) What are the eigenvalues of A
9 ?
[4] (b) What are the eigenvalues of 4 A
T
[4] (c) Write down two matrices B and C with B 6 = kC for any k with characteristic
polynomial p above.
3. Consider a simple model for determining the popularity of a website by setting
xi = 1 +
j
cij xj
where cij = 1 if website j links to website i else it is zero. In our small web:
- Website 1 links to websites 2 and 4
- Website 2 links to websites 1, 2 and 4
- Website 3 links to website 1
- Website 4 links to website 2
[4] (a) Write down a matrix C and a vector b such that the popularity vector x satisfies
x = Cx + b
[2] (b) Write the system that x solves in the form Ax = b.
[2] (c) How do you know that x = A
− 1 b exists?
[4] (d) Without solving the system explain which is the least popular website.
4. This question concerns the projection of vectors in R
2 onto the line through the origin
and parallel to v = 〈 1 , 1 〉.
[6] (a) Given that cos π/4 = sin π/4 = 1/
2 show that the standard matrix for the
projection is
A =
[
]
[6] (b) Reasoning geometrically find the eigenvectors and eigenvalues without using the
matrix A above. (HINT: draw a sketch!)
