Prof. Dancis                                              Spring, 2004


Welcome to Math 461, an introduction to Linear Algebra for students of science and engineering.


This is an introductory course in linear algebra, mainly matrix algebra.  An important goal will be to improve your problem solving abilities. 

            The textbook for this course is my book entitled:  An Introduction to Linear Algebra for Science and Engineering Students.  This book (with green cover) will be available from the university's book store.  (The university keeps all the money;  I receive no royalties.) The book is sold without a binding; only the 600+ pages with 3 holes punched; you will need to supply a wide 3 ring binder, a slant ring binder works best. 

MATLAB.  The Math Dept. has decreed that the use of the powerful Mathematical software  MATLAB, will be a part of Math 461.


No previous knowledge of MATLAB is assumed; the Math dept. expects that students will teach themselves how to use this software with the aid of the Academic Information Technology Services (AITS) course: "Introduction to MATLAB”.  AITS's Peer Training Spring 2004 schedule can be viewed at  The MatLab course usually fills quickly.  Cost is $10.  If you re not familiar or are familiar but not comfortable with MATLAB, you are urged to sign up in Room 1400 of the Computer Science building or online at


MATLAB  Help:  A Math Dept. graduate assistant will be available to assist you with MATLAB  and with problems that arise as you work on the MATLAB  assignments. 


The Math Dept.'s "resources" website has leads to much useful info.   It may be found at  This includes Computer

Resources, Tutoring Resources  and the Math Dept. Testbank of old exams.

In this course, you will become fluent in symbolic matrix algebra calculations. You will develop facility with simple proofs mainly proof-by-calculations.  You should acquire understanding and ownership of applicable linear algebra (like least square fits). The course will include training in problem solving


Approximate Syllabus


Coordinate and geometric vectors, including dot products.


Matrix Algebra and Matrix transformations and their many connections. 

Proving Matrix identities

Applications: Superposition for (electrical) D.C. resistance circuits.

Finding particular solutions to special systems of non-homogeneous linear differential

Equations (not in Math 246 (Differential Equations) syllabus).

Least square fits (with perpendicular projections) including multiple regressions and the Normal Equations. 


Solving systems of linear Algebraic Equations using the Gauss Elimination Method.

Perturbation Theory - the surprising effects of measurement errors (for matrix  M  and vector  w)

on the solution  v,  to  Mv=w.  Also how to quickly obtain good estimates.               

Design problems for  Mv=w  (How tight must the tolerances on  w  be, in order that the

specifications on  v,  will be satisfied?)

Simplified version of Computerized Tomography,


Non-standard linear coordinate systems for Rn and matrix change-of- coordinates. 

Graph equations like  (y-x) = ±(y+3x)5  using a change-of-basis matrix.

Dimension theory


General Linear equations        

Theorems of general linear equations    An emphasis will be on explaining Math 246.




Eigenvalue Theory (without Jordan form -- no double roots)  includes complex eigenvalues.


Solving finite difference equations:  vn+1 = M vn   Application to Arms Races and stochastic matrices.

(Review) the basic systems of linear differential equations  (v' = Mv)


Quadratic Forms - Max and Min

The Spectral Theorem for real symmetric matrices.


Reading the book.  This textbook was written to be read.  This text explains, in a thorough manner, how the linear algebra is used.  The text is packed with content.  You will need the textbook in order to fully learn the material.  I am aware that many high school textbooks are big on mentioning topics without explaining them.  High school texts are also short on content and on analysis of the material.  The result is that there is little value in reading many high school texts and so many students get in to the habit of not reading textbooks.  This text is the "opposite" of a high school one.  A bonus point will be awarded to the first student who informs me of a typo in the book.

Problems:  The textbook has fewer routine–type problems, but there are still enough so that you may practice the "basic skills" taught in this course.  The purpose of many problems is to indicate the variety of ways that the material (in the book) is used.  Many problems provide data or foreshadow material (like a new concept or theorem) which will appear later in the book.  An important purpose of the problems, together with the group work, is to teach you to improve your

When HW is assigned, some exercise numbers will be listed with one or more stars;  one star indicates that the exercise is a small challenge, two stars indicates that the exercise is a medium-level challenge. 

Teamwork:  Occasionally, you will be doing board work in teams.  When this is announced, stand up and organize a team of three or four students.  Introduce yourselves to each other.  Lay claim to a section of the blackboard by putting all your names on it.  Often these boardwork exercises are background or foreshadowing for the next day's lecture.

As a team work through the problems.  When someone makes a mistake like writing "3 + 4 = 8", respond in an adult manner by saying that the mathematics is incorrect or wrong and then correct it.  Do not say the person is wrong or stupid.  Do not give a high school, cutting remark, type of response.   Do not make personal remarks.

It is the responsibility of the team members to fully explain all parts of the solutions to the other members.  Learning to explain mathematics to your peers is an important aspect of both teamwork and your mathematical training.

It is also important to learn to spot mistakes in the work of others (as well as your own).  It is important to learn how to accept criticism in an adult manner––this includes how to defend your work when it is correct.  Professor Treisman has observed that:  "through the regular practice of testing their ideas on others, students will develop the skills of self–criticism essential not only for the development of mathematical sophistication, but for all intellectual growth."

Study time  You should budget  6 hours per week for study and problem solving.

The recommended method of study is:

1.  Spend 1–2 hours alone, reading the text and class notes, before working lots of problems.  The emphasis should be on understanding the mathematics not just on getting the right answer.

2.  Follow the individual study period with 1/2 – 1 hour of discussion and team learning which you discuss the mathematics and the harder problems with two or three other students.  Also critique each others work.

To encourage group work, homework may be submitted with up to four names on it.  Each person is required to proofread the final draft.  Extra points may be subtracted for errors in group homework.

Unlike the "e - d" definitions in calculus, the statements of definitions and theorems, in this course, are "reasonable".  It is important that you learn them.  On tests, you will be expected to state accurately and coherently the more important definitions and theorems.  (This need not be done verbatim.)

Checking Answers:  It is important to develop the habit of checking answers whenever possible.  Methods of checking answers will be taught.  (You will be expected to check your answer on tests.)  If an answer does not check out, read over your work, find the mistake and correct it.  Then check your new answer.  On tests, if an answer does not check out and you do not have time to find the error, write down "the answer does not check out––something is wrong".

My philosophy of education is presented in my article "A Hybrid Small-Group Guided-Discovery Method" (Page 11).  The goals of the course are presented in my composition "Welcome to Dr. Dancis’s Notes".  Notes on Selected Sections list many topics which are special to this textbook.

Arithmetic mistakes:  Mistakes in addition, multiplication, and copying will count 1 point or zero out of 10.  Exception:  If the mistake makes the problem easier, then you only get credit for the work that is done.

All other arithmetic mistakes will count at least 3 or 4 points out of 10.  

No credit   for mistakes which demonstrate serious lack of understanding such as

    1     =   1  +   1 ,      

  a+ b        a       b       

Hand calculators will not be permitted on exams.

Solutions must be clear and easy to read.  You must present work which is easy for me to read and understand.  Your solutions must demonstrate that you understand what is happening.  Having "the general idea" is not enough.  You must be able to solve the problems quickly, completely and accurately.  Work, which is sloppy, ambiguous, unorganized or incoherent will not be accepted.  I will not spend time trying to figure out what you have written.  Do not hand in incomplete homework exercises.  Place final solutions inside a box.

If you write two solutions to the same problem on a test, only the first one will be read and graded.  Points will also be taken off for unnecessary material which happens to be incorrect.  Cross out anything that you do not want to be graded.

There will be many quizzes, usually consisting of quickie one-two minute questions. Quizzes will be given at the beginning of the period. 

Homework will be spot checked; some problems graded, many others ungraded.  Drop homework on my desk at the beginning of class.

During the first two weeks, homework will be corrected, but not graded; just scored as done.

Copies of my old exams are available at the Math Dept.’s Testbank, on the web:

You may drop questions on my desk at the beginning of class.  If you are having trouble with a problem; show your attempt and put a box around the line where you are stuck or are having difficulty.  Better yet, e-mail this information to me, the day before. 

My office is Room 4419 in the Math building.   My e-mail address is Office hours are MW at 11  and MWF at 3.  It is best if you catch me at the end of class and inform me that you will be coming and/or make an appointment.

The tentative exam schedule is Feb. 23, March 31 and April 26.  Also there will be a two–hour final exam.


Your grades will be averaged as follows:

TE = test average,       FE = final exam score and

HW = score for homework, quizzes and boardwork.

Grade = 50% FE + 40% TE + 10% HW, when FE > TE

Grade = 40% FE + 50 % TE + 10% HW, when FE < TE

We will use a 90 80 70 60% scale for A B C D.



Computerized Tomography is described in the appendix to Ch. 4 (at end of Ch. 4).  At some point, read it twice.  Due.  March 29, a two page mostly-typed summary.  Due April 9  Exer. A.16




Homework – Due Wednesday, Jan. 28:

Fill out an index card.

Buy the textbook. 

Read Pages 5-26, they explain my teaching philosophy and the textbook.

Read the Prologue, pages 27-32; practice on all exercises.

Read Pages 33-36

Read Ch.1 Sec. 1, twice.  Practice on all exercises.

Any questions on this material?  List what is confusing.  List any typos that you spot.

Hand-in: Exercise 1.24, 1.38, 1.39,  1.52*. 1.55.   What about the answers is interesting? 

Remember to use “for all” notation, to check answers and to place final solutions inside a box.

Browse the rest of Ch. 1.

Friday  Hand-in: Ch.1 Sec. 1    Exer.s 1.56, 1.61 and 1.62

If you have not taken a course in differential equations, then hand-in Exer. II.1.#4,5,6*,9

Read Appendix D (Geometric Vectors) and Ch. 1  Sec. 4 and its additional examples.  

Hand in Appendix D Prob. 3.27, 28a,  and Exer. 32, 33,

Use MATLAB  to redo  Exercise 1.24, 1.38, 1.39,  1.52*, 1.55. After doing a MATLAB, calculation, rewrite solutions in easy to read form, using “for all” notation; do check answers, and do comment on answers.  Are the answers the same as when you did them by hand?   Those who have signed-up for a MATLAB class should hand-in these exercises after you have taken the MATLAB class.

Challenge exercises (optional): 34*  Also Exer.  3.24, 3.25*, 3.26*** 


Name    rank (jr. or senior or grad)      major

List the math courses you have completed in college (Courses taken at UMCP may be listed by number; courses taken elsewhere should be listed by the names of courses and college.)

How do you feel about math?  Which math course did you like most?  Why?

How do you feel about physics?                       

Why are you taking this course?

MATLAB   If no experience with MATLAB, for which MATLAB class did you signed up?

Write a paragraph on your experience, with MATLAB.  Good vs. Bad; why?  Comfortable vs. Uncomfortable --  What were the difficulties?   In which courses did you use MATLAB?

If no experience with MATLAB, but experience with another Math software package, write a paragraph about it.

Have you used matrices in other course?  If so, write at least a paragraph on what you learned and how you used matrices.  State what course(s) this occurred in.

For how many credit hours are you registered this semester?  How many hours a week do you normally spend on extracurricular activities (work, sports, clubs, commuting, etc.)?

How many hours a week (on average) did you study for your last calculus course?

Warm-up Exercises  for Math 461



Problem #1.  Our team is designing a "widget" for a space ship.  The boss says that we must calculate the length  w  of the widget  with 99% accuracy.  If it is off by more than 1%, the space ship may blow up.   The proper length  L  of the widget is connected to the height  h  of a gadget by these equations:            


                                          L +10u                               = 11            


                                                     u +10x                   = 11


                                                              x +10y          = 11


                                                                       y +10z  = 11


                                                                                 z = h = 1.


We measured  "h" as carefully as we could and found it to be 1.00, that is:  .99 <  h < 1.01.


So  "h"  was measured with  99% accuracy or error of less than 1%.  State what accuracy you predict for  L?




Now suppose Mr. Wiseguy said that we should send the gadget to the Monopoly Measuring Company which would use its special electronic equipment to measured  "h"  with accuracy of 99.99%    The Monopoly Measuring Company charges $10,000.  Would this $10,000 be well spent or wasted?      


To find out, calculate  L  again, this time using The Monopoly Measuring Company's measurement of  h = 1.001.


Comment on answers.


Problem #2.    Find the two points where the sphere           x2 + y2 + z2  = 6

meets the line:          x + 2y  – 3z  =  1

                                  y    z  =  1


Problem #3.   Find as many solutions as you can for these two simultaneous equations:

                                    w + 2x + 3y + 4z  =  10

                                                    2y + 2z  =  4