MTH 513  Fall 2006 Linear
Algebra Meets MW:
4:305:45 – Independence Hall 203 Instructor:
Dr. Nancy Eaton Email me: eaton@math.uri.edu Or by Apt MTW
8:009:00AM


Students who require
accommodations and who have documentation from
Disability Services (8742098) should make
arrangements with me as soon as possible.


Course Content
Principle topics
of linear algebra are covered. Vector
spaces are emphasized along with linear transformations and their relationship
to matrices. Topics are selected from
the following list
1. Vector Spaces.
Ø
Introduction. Vector Spaces.
Subspaces. Linear Combinations and Systems of Linear Equations. Linear
Dependence and Linear Independence. Bases and Dimension. Maximal Linearly
Independent Subsets.
2. Linear Transformations and Matrices.
Ø
Linear Transformations, Null
Spaces, and Ranges. The Matrix Representation of a Linear Transformation.
Composition of Linear Transformations and Matrix Multiplication. Invertibility
and Isomorphisms. The Change of Coordinate Matrix. Dual Spaces. Homogeneous
Linear Differential Equations with Constant Coefficients.
3. Elementary Matrix Operations and Systems of Linear Equations.
Ø
Elementary Matrix Operations and
Elementary Matrices. The Rank of a Matrix and Matrix Inverses. Systems of
Linear Equations—Theoretical Aspects. Systems of Linear Equations—Computational
Aspects.
4. Determinants.
Ø
Determinants of Order 2.
Determinants of Order n. Properties of Determinants. Summary—Important
Facts about Determinants. A Characterization of the Determinant.
5. Diagonalization.
Ø
Eigenvalues and Eigenvectors.
Diagonalizability. Matrix Limits and Markov Chains. Invariant Subspaces and the
CayleyHamilton Theorem.
6. Inner Product Spaces.
Ø
Inner Products and Norms. The
GramSchmidt Orthogonalization Process and Orthogonal Complements. The Adjoint
of a Linear Operator.
7. Canonical Forms.
Ø
The
Calculation of Grade:
Homework  25%
Portfolio  25%
Midterm exam (W 10/25)  25%
Final exam (M 12/18)  25%
Exams
There will be a
Midterm Exam and an inclass Final. Your Midterm Exam will be on Wed, Oct 25 and your final will be Mon, Dec 18. The exams will test your understanding of the
material we cover in class. You will
need to recall the theorems that we covered and understand their significance,
answer questions in general about implications of these theorems. Before each
test, I will provide you with a list of theorems from class that I would expect
you to be able to prove.
Homework Assignments:
Problem sets from the book will be assigned
throughout the semester. The book
includes some exercises for working out examples and some statements to
prove. You do not need to hand in the
examples – just check your answers in the back of the book. For the theory questions, which require a
proof, you are expected to write the solutions up carefully and hand them in by
the due dates given. All assignments
will be posted here as we go.
Assignments
Section Think about Hand
In Due date Goal 
1.1 2a,
3a 
1.2 1, 7, 10,
12, 13, 14, 15, 17, 20
19, 22 Mon Sept 18 8 pts 
1.3 1, 6, 8,
11, 12, 20, 21, 31
10, 18, 28, 30 Mon
Sept 18 12 pts 
1.4
1, 21, 2c, 5a, 5c, 6, 7, 8, 9 10, 12, 15, 16 Mon Sept 25 12 pts 
1.5 1, 2, 3, 4, 5, 6, 10, 11,
15 9, 12, 17 Wed Sept 27 8 pts 
1.6 2c, 2e,
7, 9, 10a, 10c, 12, 16, 17, 29
8, 15, 28, 33, 34a Wed Oct
11 12 pts 
Worth 2 pts 2.1: 3, 6,
11, 15 
Worth 4 pts 13, 14, 21, 24, 28, 29 
Due by: Oct 16 
Goal 12 pts 
2.2: 2b, 2e, 4 2.3: 3b, 4b, 9 2.4: 4, 5,
14 2.6: 3, 4,
5, Ex 5 
8, 11, 12, 13, 16 13, 18 6, 9, 10, 15 9, 11, 14, 18 
Oct 18 Oct 23 Nov 1  
12pts 8 pts 8 pts 0 pts 
4.3 and 5.1: 5.2: 3(a) 3(c) 
4.3: 25, 27  5.1: 14, 16 8, 9, 11 
Nov 6 Nov 13 
12 pts 12pts 
Section 5.4: 
Write out definitions and proofs as if you were going to present
it to the class. Defn: Tin variant
subspace Thm 5.21 Thm 5.22 Thm 5.23 Thm 5.24 Defn Direct sum of matrices Thm 5.25 
Nov 20 
Worth 10pts 
6.1: 5 6.2: 2(f),
2(g), 4 6.3: 
10, 12 Prove Cor to Thm 6.11 
Nov 22 Nov 27 Nov 29 
6pts 6pts Worth 4 pts 

6.3 (22d) 6.4 (4) 6.5 (10)
6.6 (7) 6.7 (11) 7.1 (7)
7.2 (3, 4b,c) 7.3 (13) 
Dec 11 
28pts 
HkPts: Each exercise will
be graded and comments will be given.
Each is worth up to 4 HkPts.
Here are some tips in doing homework.
Try each problem on your own.
Check your answer with someone from
class.
Ask me about the ones that you still do not
completely understand.
Take notes whenever we go over a problem
in class.
Carefully write a final version of each
problem, as best you can, and keep them together in a binder.
Portfolio:
From the suggested homework
problems, I will select 10 for you to rewrite and hand in as a portfolio of
work, representing what you learned in this course. This is intended to
be a collection of your best work. This will be due on December 18th. You will have two weeks advance
notice as to which problems will be included in the portfolio. The problems
will be listed on this web page on November 27^{th}. The intention here
is that you will make sure you understand how to solve each homework problem as
we go along, in case it will be selected to be in the portfolio.
Write out the entire statement of the exercises. Write the
solution as formally as you can. Use complete sentences. Write it
as if it would be published. In fact, I will publish selected solutions
on the web! This MUST BE TYPED. It is okay to type the sentences
and leave spaces for writing equations by hand. If you have access to an
equation editor, that would be best.
See: link.
DUE: 12/18
Sec 2.1: 14; Sec 2.2: 8,
12, 16; Sec 2.3: 13; Sec 2.4:
9, 10; Sec 2.6: 14;
Sec 5.1: 16; Sec 5.2:
9
Guide to
working with others:
There are
benefits to discussing the problems with your classmates. If you become
stuck on a problem, fresh ideas from someone else might provide you with some
new angles to try. In the academic community as well as in business and
industry, people often work in teams. So, it is good to get some practice
working with others. Working with someone from class will help you to
improve your math communication skills as well.
I encourage you to work together under the following circumstances.
Begin the problem on your own and do as
much as you can.
Ask someone from the class to explain the
basic outline of a solution.
If working with someone else, sometimes
they will also ask you for your understanding of the basic outline.
Take what you learned and write out the
solution on your own, using your own words.
Never copy word for word from anyone
else’s paper. In fact it is better not to look at anyone else’s completed
written solution or to show yours to anyone else.
If you still do not completely understand
the solution, you can ask your professor to look at what you wrote and try to
clear up any parts of the solution that are not completely clear and
accurate.
The following is an excerpt from
the University Manual.
8.27.11 A student's name on any written exercise
(theme, report, notebook, paper, examination) shall be regarded as assurance
that the work is the result of the student's own thought and study, stated in
the student's own words and produced without assistance, except as quotation
marks, references and footnotes acknowledge the use of other sources of
assistance. Occasionally, students may be authorized to work jointly, but such
effort must be indicated as joint on the work submitted. Submitting the
same paper for more than one course is considered a breach of academic
integrity unless prior approval is given by the instructors.
Schedule

Mon 

Wed 

September 


6 Notes 
1.1, 1.2, 1.3 

11 Notes 
1.4, 1.5 
13 
1.6 

18 
2.1,2.2 
20 
2.3, 2.4 

25 
2.5, 2.6 
3.1, 3.2 

October 
2 
3.3 
4 
3.4, 4.1 



11 
4.2, 4.3 

16 
5.1 
18 
5.2 

23 
5.3 
25 
Mid Exam StudyGuide 

30 
5.4, 6.1 
1 
6.2 
November 
6 
6.3, 6.4 



13 
6.5 
15 
6.6 

20 
6.7 
22 
6.8 

27 
6.10 
29 
6.11 
December 
4 
7.1 
6 
7.2 

11 
7.3 



18 3pm 
Final 


See above for portfolio assignment – due 12/18
If you do the portfolio, your grade will be calculated as
follows:
Calculation of Grade:
Homework – 25pts =
minimum{25 x ( your total / 170), 25}
Portfolio – 25pts
Midterm exam – 25pts
Final exam – 25pts
Total possible pts  100
The alternate assignment is given in the Notes 5.1 through 6.2 and is due 12/18
Note – if you do the alternate assignment, your grade will be
calculated as follows:
Calculation of
Grade:
Homework – 25 pts = minimum{25 x ( your total / 170), 25}
Section 5.3, 6.8, 6.9, or 6.10 – 30 pts
Midterm exam – 25 pts
Final exam – 25 pts
Total possible pts – 105
In both cases, I will use the following scale for your grade in
this course:
A: 92pts
A: 90pts
B+: 87pts
B: 82pts
B: 80pts
C+: 77pts
C: 72pts
C or lower is
considered failing in a 500 level course