MTH 513 - Fall 2006

Linear Algebra

 

Meets MW:  4:30-5:45 – Independence Hall 203

Instructor:  Dr. Nancy Eaton

E-mail me: eaton@math.uri.edu
Phone:  874-4439
Office:  Rm. 222, Tyler Hall
Office hours: After class 5:45-6:00

Or by Apt MTW 8:00-9:00AM


Visit my web page: http://www.math.uri.edu/~eaton/

 

 

 

Students who require accommodations and who have documentation from
Disability Services (874-2098) should make arrangements with me as soon as possible. 

 

 

 

 

 

Text:  Linear Algebra (4th Edition) (Hardcover) by Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence

Check Amazon.com

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 Cayley-Hamilton Theorem.



6. Inner Product Spaces.

Ø                 Inner Products and Norms. The Gram-Schmidt Orthogonalization Process and Orthogonal Complements. The Adjoint of a Linear Operator. Normal and Self-Adjoint Operators. Unitary and Orthogonal Operators and Their Matrices. Orthogonal Projections and the Spectral Theorem. The Singular Value Decomposition and the Pseudoinverse. Bilinear and Quadratic Forms. Einstein's Special Theory of Relativity. Conditioning and the Rayleigh Quotient. The Geometry of Orthogonal Operators.



7. Canonical Forms.

Ø                 The Jordan Canonical Form I. The Jordan Canonical Form II. The Minimal Polynomial. Rational Canonical Form.

 

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 in-class 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:  T-in 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 27th. 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

27Notes transformation

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

Study Guide

 

 

Notes 5.1 through 6.2

Notes 6.3

Notes 6.4 through 7.3

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