The University of Sydney
MATH2022 Linear and Abstract Algebra
Semester 1 Assignment 2 2022
This assignment comprises four questions and is worth 5% of your final mark. It should
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introduced in lectures. Present your arguments clearly, using words of explanation that
connect formal use of symbols. After all, mathematics is about communicating your ideas.
This is a worthwhile skill which takes time and effort to master.
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1. For each of the following statements, indicate whether they are true or false. If a
statement is true, prove it. If a statement is false, give a counterexample.
Let n and m be non-negative integers and let F be a field.
(a) The cartesian product of two fields is a field with coordinatewise addition and
multiplication.
(b) Let T : F n → Fm be a linear transformation. If the matrix that represents T has
rank n, then T is injective.
(c) Let V be a vector space over F . The span of any two distinct nonzero vectors in
V has dimension two.
(d) Any three nonzero polynomials of distinct degrees are linearly independent in P10.
2. Let n be a positive integer and F a field. Recall that we write GL2(F ) for the group
of invertible 2× 2 matrices over F under matrix multiplication. Consider the set
H =
{[
1 a
0 1
]
∈ Mat2(F ) | a ∈ F
}
.
(a) Prove that H is a subgroup of GL2(F ).
(b) Is the group H abelian?
(c) Prove that the group H is isomorphic to the group (F,+).
3. This question refers to this research paper. In this paper, the authors study a phase
field model for two-phase flow in a capillary tube (this is important for applications like
oil recovery, ground-water flow with contaminants, and carbon sequestration, as well
as to model the flow of fluid in lungs). In particular, they find a bound on parameters
to obtain physically relevant solutions for the differential equations that appear in this
model.
In section 2.2 of the research paper, the authors study a system of differential equations
with unknowns u, v and w. As we have seen in lecture, the nature of the solutions of
the system will depend on what the eigenvalues of the coeffient matrix look like. The
coefficients in the system depend on certain parametersM, Ca,C1, C2 and s. You may
use that MCa
C2
(s− 1) is a positive real number.
(a) Verify that the eigenvalues λ1, λ2 and λ3 of the coefficient matrix are the roots of
the function given in (6).
(b) Explain why it follows that λ1λ2λ3 is a positive real number and λ1+λ2+λ3 = 0.
(c) The authors conclude that one of the eigenvalues is a positive real number, and
that the other two are complex numbers with negative real parts. Why is this
true? Hint: for this part, you may us the complex conjugate root theorem.
4. In the 1990’s, internet searching was very inefficient. Search engines like AltaVista or
Yahoo would scan pages and keywords for the word you were looking for, and roughly
list the results with the most occurrences of this word. Web pages could exploit this
by populating web pages with invisible keywords so they would be ranked first in
the search results. In 1995, Larry Page and Sergey Brin started collaborating on an
algorithm (nicknamed BackRub) to rank web pages by importance. They published a
research paper and founded Google based on this algorithm.
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In this algorithm, every page has a PageRank, or an ‘objective measure of its citation
importance that corresponds well with people’s subjective idea of importance’. In this
exercise, we learn how it works.
Google’s way to measure importance (a positive real number), is by the following rule:
Write C(A) for the number of links going out of a page A. If page A links
to page B, then B inherits 1
C(A)
of A’s importance.
To illustrate, suppose my web page is one of a gazillion fan pages for the Spice Girls.
Write P for the official Spice Girls web page.
ˆ All fan pages link to the official page P , so P inherits some importance from (at
least) a gazillion pages. Hence the total importance of the page P is very high.
That is, the page P is very important.
ˆ Suppose the official page P (or some other important site) links to a few fan
pages, including mine. Then my page should also be considered important.
The more pages that P links to, the less importance my page inherits.
ˆ On the other hand, if almost no pages link to mine, and the ones that do are not
important, then my page is also not important.
Suppose the internet has n pages P1, . . . , Pn and that C(Pj) > 0 for all j ∈ {1, . . . , n}
(this means that every page contains at least one link). Write M for the n× n matrix
whose i, j-entry is 1
C(Pj)
if Pj links to Pi, and zero otherwise. Write v for the column
vector of length n whose i-entry is the importance of the page Pi.
(a) Explain why M is a stochastic matrix.
(b) Show that v is an eigenvector for M with eigenvalue 1.
By convention, we require that the importance of all the pages in the internet adds up
to 1. By part (b), v is a steady state probability vector for M . We would like to use
this observation to find the vector v, and hence the importance of each web page. But
M may not be a regular stochastic matrix, so the steady state probability vector may
not be unique! Google’s solution is to replace the matrixM by theGoogle matrix G:
G = (1− d)M + dN,
where N is the n×n matrix whose entries are all 1
n
, and d is a parameter which Google
sets to 0.15.
The Google matrix G is a positive stochastic matrix. The PageRank vector PR is the
(unique) steady state probability vector of the Google matrix. Now, the PageRank of
page Pi in Google is the i-entry of the vector PR.
(c) For simplicity, imagine an internet with only three pages: page A links to pages
B and C, page B links only to page C, and page C links only to page A.
Compute the Google matrix G for this internet.
(d) The Google matrix G for the internet in part (c) has steady state probability
vector PR ≃
0.3880.235
0.397
 (you do not need to verify this). Which page would Google
rank as the most important?
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