Math 308 Introduction to Linear Algebra
(Fall 2011)
Thursday December 8.
Another
old final.
CLUE will have a review session for Math 308 from
8 pm to 10 pm in Mary Gates Hall Room 295
December 6.
I made some small changes to the sections of
my notes
we have been discussing recently.
November 28.
Eigenvalues and eigenvectors.
I made some small changes to the section on determinants in
my notes.
November 23.
Determinants.`
November 21.
More linear transformations.
Chelsea Walton's lecture
notes.
November 18.
An important new topic today, linear transformations. Before coming
to class read as much as you can from Chapter 12 in my notes,
but ignore the
material from section 12.11 onwards.
Section 3.7 of the book also introduces linear transformations.
Chelsea Walton has kindly offered to teach for me on
Nov. 18 and 21. Her web page is here.
Chelsea Walton's lecture
notes.
November 16.
Check the homework due on Monday at the bottom of this page.
Oh, let's make it due on Wednesday next week.
November 16.
Today's class covers section 3.6 in the book BUT I do not expect
you to understand the Gram-Schmidt process which shows you how to
construct an orthonormal basis from an arbitrary basis. It is good
to know there is such a process but the details of it can be
omitted. Read section 11.10 in my notes.
Notice that it begins with a generalization of the dot product
which allows us to talk about vectors being perpendicular to one
another. Read and understand the proofs of Lemma 11.17 and
Proposition 11.18 in my notes. For an important class of
orthogonal matrices look at section 12.6 and check that the columns
of the matrix in (12-2) give an orthonormal basis for R^2.
November 14.
Take a look at the proof of Corollary 11.11 which proves that
rank(A) = dim(range of A). This is an important result. Many
books define the rank of a matrix as the dimension of its range.
We have already seen that the range is the linear span of the
columns (use the wonderful formula for Ax) so Corollary 11.11 is
really about the relation between the dimensions of the row and
column spaces of A. See Corollary 11.13 for a precise statement:
the row and column space have the same dimension. Read section
11.9.
November 15.
I have updated
my course notes today.
To check whether you have the latest version look at page 66: it
should begin 12.1.4 How do we tell....
If your version doesn't have that try clearing your cache and
reloading the page.
Solutions to the Midterm:
These
solutions and comments were posted on November 13.
Important dates:
The Midterm for both sections will be on Friday November 4.
The final exam for Math 308I is Thursday December 15 at 8:30 a.m.
The final exam for Math 308J is Monday December 12 at 2:30 p.m.
You will need a bubblesheet, and either a blue book or green book for the
exams.
Office Hours:
Wednesday 2:35-4:30 in Padelford C-418,
and by appointment.
I often find office hours the most enjoyable part of the course. I
get a chance to know you, you get a chance to know me. My job is to
help you learn this material. You are often likely to feel lost and
perhaps stupid. That is normal. I often feel lost and stupid when I
am working in a new area of math. It is part of the process. But I
can provide you with guidance and help. If I can stretch some
medical metaphors (my wife is a physician), it is no good coming
to see the doctor or start taking the medicine once the illness is
terminal. Catch it in the early stages, at the first sign of a
sneeze or ache, and the chances for recovery are good. (I am not as
fearsome as I might appear---remember that I was once your age,
feeling small on a big campus, and that I have experienced and
continue to experience the struggles
we all have as part of life. And, heck, the math struggles
are
among the smaller ones! So, why not start praticing in the small
arena of math to say "I need help".)
Textbook and Course Notes.
The textbook is ``Introduction to Linear Algebra" by
Johnson, Riess, Arnold; 5th edition.
We will cover Chapters 1, 3, and 4. You are expected to be familiar
with the material in Chapter 2---you will have seen in it in Math
126 or an equivalent course on multivariable calculus.
My
notes for the course are an additional ``textbook''.
If you find typos in my notes I would be very happy to hear about
them. If you have suggestions for improvement or
would like me to explain a little more about some topic please let
me know. Thanks!
Definitions.
Definitions are the foundation on which mathematics is built.
There are many definitions in this course. More than in any course
you have previously had. If you don't know the definitions
forwards, backwards, and sideways, you will certainly fail this
course. Start learning them the first day of class. Build your own
list of definitions. Review them regularly.
One reason students find this course so difficult is that they
don't make a real effort to understand the definitions. Or, they
leave that affort too late. You must begin the first week of the
course and constantly monitor yourself. You can monitor your
progress by being honest with yourself when I write things on the
board. If you don't understand a word I use, ask. There will be
others in the class who don't understand it either. By asking you
do everyone a favor. Even many of those who think they understand a
definition don't.
The document
definitions consists of my response to an email from a student
who is asking whether he has the correct definitions for some of
the terms that arise in Linear Algebra. Mostly the student is close
but not close enough---all it takes to sink a boat is one hole, and
the same holds for definitions: either it is correct or it is
wrong.
I always ask for a lot of definitions in my exams
because they are so important. If you don't know the definitions
you will surely fail this course.
Here is another note about definitions and comments on various
errors made by students.
Set notation and language.
We will use the basic language of sets and functions in
this course so I have written some
notes about what you need to
know.
You can take an online test to see how well you
understand some of the ideas.
Grading.
Your grade will be based on the homework, the midterm, and the
final. Your homework scores will contribute 10%, the midterm
will contribute 20%, and the final will contribute 70%.
Pop Quizzes.
There will be unannounced 10-15 minute quizzes every 1 or 2 weeks
(depending on my energy level and free time).
Here are some examples.
The Midterm.
The Midterm will cover everything we have covered in class so far.
I encourage you to read the course notes thoroughly, also my notes
on
set-theoretic language and
definitions. I always ask for a lot of definitions in my exams.
I have posted some old midterms some of which include answers, and comments on the questions, answers, and common errors.
Here is some good advice: print off all these
old midterms at the beginning of the quarter and study them assiduously.
Each week look at the questions that relate to the topics
covered that week. See if you can do the questions.
Check that you understand my answers and comments.
Ask questions if you don't understand.
Ignore this advice at your peril.
2003 midterm with answers and comments on incorrect answers.
2009
midterm with answers and comments.
Another 2009 midterm with answers and comments.
On this midterm, most people did well with the definitions.
The True/False questions were a killer though.
Part A revealed a lot of weaknesses that were not revealed by the
other questions.
The Final targeted some of the areas of weakness so contained
variations on the following midterm questions: Part A:
2,3,4,5,6,7,8; Part C: 1,4,6,7,10,11.)
Fall 2010 midterm with answers.
An extra long practice midterm.
2011
Midterm with some answers
I want you to know what I want you to know.
I want you to be able
to answer the questions on these midterms.
At least 75% of the questions on your midterm will be the same
as those on these old midterms.
Final.
Here is an old
practice final.
It is far longer than the real one
will be, but you will get a sense of your abilities by trying it.
And another
old final.
I can not overemphasize the importance of going through the
midterm with my answers and comments in detail. You should then try
to retake the midterm under test conditions and see how well you
do. I would not be surprised if you get close to 100%, if not on the
first attempt, maybe on the second. You should do the same with the
final.
Other important information.
As you will see there are some multiple choice questions
and some True/False questions. You will need a purple ``Standard
Answer Sheet'', which you can buy at the bookstore in the Hub, or
on the Ave, or at the Newsstand in the Hub, and at other places
around campus. You will need a Number 2 Pencil to fill in the
bubbles on the Answer Sheet. You must also put yuor section and
your student ID number on it. Make sure you come to the exam
knowing those things.
How to succeed.
You want a good grade and I want to give you a good grade.
I will give you a good grade if you
demonstrate some degree of mastery of elementary linear algebra.
Learning math requires much more than reading books, or re-reading
the notes you take in class; that is necessary of course, but not
sufficient. You learn math by solving problems, doing exercises,
both those I assign, and those you find yourself in the text book,
and even in other books on linear algebra.
You will not master the material in this
course if you do only those homework problems that I assign.
This
is a truth, a fact. Solving problems is the only way to
learn mathematics. So, do hundreds during the eight weeks of this
course. Yes, hundreds. I understand that your time is precious, and
that there are many demands on your time. But that cannot change
the fact, the reality, that to master 3 hours of lecture material
you probably need to do on the order of fifty problems; not just
fifty variations on the same problem, but fifty different
problems.
It might be helpful for you to study with others. Check each others
solutions to problems, talk about the ideas and results.
Homework.
Each week I will give you a list of about 20 problems for Homework.
Of those, three will be graded. Each problem will be graded out of
three points. You will also receive 10 points simply for doing all the
homework problems and turning in neat, legible,
easy-to-understand work. Thus each week there is a total of 19 points
awarded. Really excellent work will receive a 20.
Homework is due at the start of class the day it is due.
Except in extraordinary circumstances, late homework will
not be accepted. If you miss the homework deadline, that homework
will be one of those which does not contribute to your best six.
I expect all homework to be neat and legible. Part of the homework
is to make your solutions easy to understand. Write neatly. Space
out your work. If you have more than one page for your solutions
staple them together. Unstapled homework solutions will not be
accepted. The 10 points you get for just doing all the
problems will also involve whether your homework is neat and tidy
and legible. So, even if you do all problems but it looks a mess or
is difficult to understand you will be awarded less than the full
10 points. For example, if the paper you hand in is tattered down
the left-hand side you wil receive less than 10 points.
Part of your job is to make the grader's life easy by
handing in work that is legible and easy to understand.
You should check the grader's arithmetic. If you think there is an
error in the grading return the homework to me with a written
explanation of your complaint. I will return it to the grader.
If that does not resolve the problem, then they can come to you.
What follows is a list of Homework exercises and
due dates. I have kept the comments made by the grader for a
previous Math 308 course I taught. They might be helpful.
Homework 1:
due October 7
Section 1.1: 17, 29, 32, 35....
Section 1.2: 10, 21, 31, 35, 39, 49, 51....
Section 1.3: 4, 12, 18, 23, 24, 33....
Section 1.4: 3, 8, 9....
The grader's comments:
Problems 51 and 8 were the killer ones for many people. There were
lots of computational errors on other problems as well. One of
the main reasons why students had lots of errors is because they
perform too many operations at a time using the same row.
Several students have problems understanding the ``consistency
condition'' (see page 6 for the definition, page 19 for a
discussion of how to recognize an inconsistent system, and section 1.3
for more details).
There were a few who gave answers but no work; they got zeros.
Based on their performance on this homework, I think quite a few
students need to improve their elementary row operation techniques.
There are a lot of errors made on row reducing a matrix to its
echelon form. On # 23, and 24, I was surprised to see the number of
students that had problems reducing the augmented matrix. Some
students are confused with the definition of a non-trivial
solution, and whether a system is consistent or inconsistent. On
# 33, a lot of the students couldn't solve the equation correctly,
# mainly due to the difficulty of reducing the augmented matrix.
On problem 49 in section 1.2, most people had no problem
constructing the matrix and row-reducing to find the correct
solution. However, there were some who didn't use matrices at all,
instead choosing to independently manipulate the equations in the
system to find the solution. While this still gets you to the same
place, the purpose of these early sections is to get students
comfortable with row reduction. You need to practice using the
tools the section gives you or you won't learn. Trust me, you need
to be able to utilize matrices properly or you WILL struggle later
on.
Problem 18 from 1.3 tripped a few people up, but not too many.
Most of the students who got it wrong didn't pay attention to the
fact that the system was homogeneous, meaning it also has the
trivial solution. They claimed it only had the unique solution the
book gives. Many of those who answered correctly, however, overdid
their answer. While there's nothing wrong with justification,
taking up half a page to explain why there are infinitely many
solutions can be considered excessive.
The biggest surprise on problem 24 was that many students entirely
forgot part b! They need to pay attention to what the problem
actually is. The only mistakes besides that, really, were basic
mistakes with row reduction. Students should double-check their
operations, especially in these early stages.
Homework 2:
due October 17
Section 1.5: 6, 8c, 10c, 12b, 20, 24, 30, 40, 48, 52, 56,....
Section 1.6: 26, 27, 28, 40, 43, 44, 46
The grader's comments:
Be careful when
copying down the statement of each homework exercise.
Write grammatically and mathematically correct statements. Just using
equal signs would be a step forward. Use nice paper and staple the pages
together. Some people would do better if they broke each problem into
smaller steps.
Most students know that in #27, A is not equal to B, because
cancelation doesn't work on Matrix, and there's no division in
Matrix, but some of them didn't give a counter example. On #28, a
lot of the students have trouble telling the flaw: "the argument
assumes that if the product of two matrices is zero, then one of
the factors must be zero". On #46, many students don't know the
purpose of this problem is using Theorem 10, that (A+B)^T=
A^T +B^T.
Almost everyone did #52 from 1.5 correctly. For #44 sec 1.6, most
students had trouble step (c). They either forgot to mention about
using property 2, addition is associative or didn't substitution
A+C by B+C. #46 is the worst problem. A lot of students didn't say
(x-y)T= xT- yT, and xTy= yTx.
Overall, the performance on this homework is better than last
weeks.
1.5 #52: Once again, students must remember to continue past part
a of a problem--many stopped there, thinking they were done. There
were few errors for those who completed the problem, however.
1.5 #56: The goal of showing that the given matrices work as
solutions is to give you a "hint" as to what the answer is. The
matrix multiplication they use is not the point of the problem.
Many multiplied the matrices out but then failed to explain why the
equation had infinitely many solutions.
1.6 #26: This problem was fairly straightforward as far as the
issues students were having go. There were those who remembered
that not all matrices A and B are commutative, and those who
didn't. Those who did got the problem right, and those who didn't
got it wrong.
Homework 3:
due November 21
Section 3.4: 1, 9b, 9c, 11, 24, 26c, 34, 37, ....
Section 3.5: 4, 6, 9, 18, 26, 27a, 32, 33c, 34, 37, 38.
The grader's comments:
For 3.4 #24, part a) some students had problems reducing the
matrix formed by the vectors in S. They also need to point out
that x_4 is an unconstrained variable. so the basis is {S1,S2,S3}.
part b) some students forgot to take the transpose of the final
reduced form.
For #37, a), most students know that any set of 3 or more vectors
in R^2 is linearly dependent, and is not a basis for R^2.
b) some students don't know how to obtain a
contradiction by using the "Hint"- {e_1, e_2} is a
linearly independent set. if a basis has only one
vector, then e_1 and e_2 are dependent.
For 3.5#18, by far, this one has the highest correct
rate! Almost no one had trouble with this one.
Everyone did a good job, except a few students
didn't show any work.
For 3.4 #24, most students did well. part a), it's important that
students show x_4 is the independent variable, the desired basis
is
{u_1,u_2,u_3}. I took off 2 pts for some students who didn't do part
b)
For 3.5 # 37, I didn't see many students did it the way the
solution
manual does. For most students, I took off 2 points if they didn't
say
rank(A)= rank([AIb]), and use Theorem 11 to claim that the system
Ax=b
is consistent. Since I saw so many students did it in different
ways,
you might need to make some clarifications on this one.
Homework 4:
due November 28
Section 3.6: 4, 6, 10, 16, 20, 24, 25, 28, ....
Section 3.7: 2a, 3c, 3d, 4, 10, 18,
19a, 19c, 21, 22, 24, 28, 30, 34, 36, 44, ...
Supplementary Exercises at the end of Section 3: 6, 10, 13....
Section 3.8: 1, 2, 3, 4, 10, 14, ....
Homework 5:
due December 5
Section 3.9: 8, 16,....
The grader's comments:
For 3.6#16,
students know pretty well about using the Gram-Schmidt process to
generate orthogonal set. The few mistakes are due to
mis-calculation. 3.7#18, most students had e_1 is an orthonormal
basis for W. Some didn't know how to use equation 4 of example 3:
T(v)= (v^T e_1)e_1= [a, 0, ]T. #28 , i don't think anyone had problem
figure out A. rank=3, nullity=0.
3.6#25, need to compare
||x+y||^2 and (||x||+||y||)^2. some students expanded
(||x||+||Y||)^2 wrong. 3.7#21, I gave only 1-2 points if
students don't show the work on how they obtained T(x). I saw
other ways of doing it besides the solution manual method, might
need some clarification on this one.
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