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 http://www.oit.umd.edu/pt/schedule.html. 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 http://www.oit.umd.edu/units/as/pt/registration.html.
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 http://www.math.umd.edu/undergraduate/resources. 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
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.
Determinants
Eigenvalue Theory (without Jordan
form -- no double roots) includes
complex eigenvalues.
Diagonalization
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: http://db.math.umd.edu/testbank/.
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 jdancis@math.umd.edu. 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***
INDEX
CARDS
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