SECTION INSTRUCTORS OFFICE TEL.
L0101
R.
Stanczak
NC
68A
416-946-8435
L0201
H. Li
FI 436
416-348-9710ext.4036
L5101
P.
Greiner
SS4087
416-978-4174
Course Administrator
R. Stanczak, office hours: Mon.3-4,
Tue.3:30-4:30, Fri.3-4.
Home page
http://ccnet.utoronto.ca/20051/mat224h1s
Textbook
W.K. Nicholson: Linear Algebra with Applications , 4th edition.
Supplementary Texts
W.K. Nicholson: Partial Solution Manual (for 4th edition)
R. Stanczak: Linear Algebra II, Study Manual -available at
DISCOUNT
TEXTBOOKS ,
229 College St.
Brief Description
This is the second course in Linear Algebra, that is more theoretical
in nature
then your
first course (Linear Algebra I - MAT223). The course will cover:
abstract
vector spaces,
linear mappings, linear operators on both real and
complex vector
spaces,
inner product spaces, orthogonal (unitary)
diagonalization of linear operators,
isometries.
It will be assumed that you know basic material from Linear Algebra
I,
particularly: matrix
arithmetic, similarity and diagonalization of matrices and
the basic
concepts of the n-space R^n
including orthogonality.
Tutorials
Every student must be registered in one tutorial section. You may
register
in one
of the
tutorial time slots through ROSI before the end of the second week
of
classes.
Registration or any changes to your tutorial time after the second
week of classes must
be done in person through R. Stanczak during his
office hours.
By the end of the third week of classes you will be enroled in one
of the tutorial
sections.
Once your designated tutorial has been posted on the bulletin board
(south-east part of the main floor of Sidney Smith Hall) and on the web
site, you
will not be allowed to switch sections.
Tutorials start at the beginning of the 4-th week of classes. During
your tutorials
the T.A.will
discuss with you some problems from the list below. Feel free to ask
questions about the
problems you have most difficulty with.
There will be 3 quizzes , typically 40 minutes long, each graded
out of 20.
Questions given on quizzes will be given out of the list of
suggested problems.
In questions where only calculations are required,
numbers may be changed.
The best two out of three
quizzes
will determine yourtutorial mark.
You must write your quizzes in the tutorial section in which you are
registered,
or your mark
will be recorded as zero. There will be no make up quizzes.
Midterm test
There will be one 110 min midterm test common to all students, during
the
seventh week of
classes. Make sure to see the schedule below for the exact
day/time.
If you have a time
conflict, contact the course administrator as
soon as possible. The
place in which you will
write your test will be advertised
two weeks in advance. There will
be no make up tests.
The test will be graded out of 60 and will consist of two equally
weighted
parts.
The first part
will have 10 simple multiple choice questions in which only your
final
answer will count.
The second part will consist of 4-5 written questions
in which
you have to present your solutionin full. In the second part, your
mark
will depend on the clarity of your presentation and, of course,on the
correctness
of your solution.
Remarking Procedure
Your test/quiz will be returned to you in the tutorial
section in which you are
REGISTERED
as soon as they are marked, usually within one week.
If you have any
questions about the
marking of the second part of your test,
you will have
to return your test to your T.A. within
10 min indicating on the
front page which question you want to be
remarked.
If you take your
paper with you, no one will look at it again. The same procedure
applies
to
quizzes.
You should collect your test/quiz submitted for remarking from the
course
administrator within three weeks, otherwise your mark cannot be
changed.
No test will be remarked unless the original answers were written in
ink.
The answer sheet of the first part of your test will be removed
from your paper
so you are
advised to mark your answers by each question also. Contact the
course
administrator directly
if you want to see your answer sheet anyway.
Missed Term Work
If you miss the test/quiz for a legitimate reason which you can
document
( i.e Doctor's note etc) your grading scheme will be adjusted by
increasing
the final exam component of your mark. Basically the mark from your
final
exam will replace the "0" on the respective missed piece
of term work. But under no cicumstances can the final
exam count for more
then 80% of your
final mark. The proof (like doctors note) should be submitted
to the
course administrator not
later then 7 days after the test/quiz was written.
Grading Scheme
The format of the final exam will be similar to the format of the
midterm
test and
will be
graded out of 100. Your final grade will be determined as follows
Tutorials
(quizzes)
- 20%
Midterm
Test
- 30%
Final
Exam
- 50%
SCHEDULE AND SUGGESTED PROBLEMS
You should solve, at the very minimum, the problems on the list below.
To
prepare well for
the test and the final exam you should solve the problems
from the
Linear Algebra II , Study Manual.
Your instructor may be slightly ahead or behind this schedule, but
the topics
covered
on the quizzesand the test will be the same for all students.
Letter S below refers to the Study Manual.
WEEK 1 (beginning Jan.03) Lecture: Complex
n-space C^n and
its inner
product.
Complex Matrices
App. A (textbook) #3(b), 3(f), 4(d), 5(b), 9, 11(d), 14
7.6 # 1(d), 2(d), 3, 4(b), 4(d), 5(b), 5(d), 5(f), 6(b), 6(f),
7, 8, 9, 11, 12, 13, 15,
18, 19, 22
Note: Skip the proofs of Th.7 and Th.9 as we will be able
to give more elegant
proof at
the end of
this course.
WEEK 2 (beginning Jan.10) Lecture: Hermitian
Operators,
Orthogonal
diagonalization
7.2 # 1(f), 2, 4, 5(d), 5(f), 6, 7, 10, 11, 12, 14, 15, 17, 18,
23
Note: All results for symmetric matrices will be introduced
as a consequence of
the
properties of Hermitian matrices.
WEEK 3 (beginning Jan.17) Lecture: Quadratic forms,
positive definite forms.
7.7 # 1(d), 2(b), 2(d), 2(f)/Q2- diagonalization only/, 3(b),
3(d), 9(a)
7.3 # 2, 3, 4, 5, 7, 9, 10
Note: In section 7.7 skip the subsection "Congruence". Section 7.3
is
covered
only
up to Example
1. The test: det^rA > 0, for positive definite forms
will
be
adopted
without the proof.
WEEK 4 (beginning Jan.24,
tutorials start) Lecture: Vector
Spaces and Subspaces
6.1 # 1, 2, 3, 4, 6, 8, 10, 11, 12, 15, 16
6.2 # 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16,
20, 27
WEEK 5 (beginning Jan.31, QUIZ#1-covers 7.2, 7.3, 7.6,
7.7) Lecture:
The Dimension
Theory
6.3 # 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17,
19, 21, 23, 24, 28, 30,
32, 35
6.4 # 1, 2, 3, 4, 5, 6,7, 8, 9, 11, 12, 13, 17, 19(a), 22
WEEK 6 (beginning Feb.07) Lecture:
Linear Transformations
8.1 # 1, 2, 3, 4, 5, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21
8.2 # 1(b), 1(d), 2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14,
15, 17, 20
R E A D I N G W E E K
WEEK 7 (beginning Feb.21) Lecture: Isomorphism
Matrix
Representation of Linear Transformations
8.3 # 1, 2, 4, 5, 6, 7, 9, 10, 12, 13, 14, 15, 23, 26
8.4 #1, 2, 3, 4, 5(d), 6, 7(d), 8, 9, 12, 16(a), 17, 18(a),
19, 22
MIDTERM TEST, February 24, 6 - 8 p.m. Covers: 7.2-3, 7.6-7, 6.1-4, 8.1-2
WEEK 8 (beginning Feb.28) Lecture: Change of Basis,
Invariant Subspaces.
8.5 # 1, 2, 3, 4(b), 5(a), 6, 7, 8(a), 9(d), 9(f), 10, 11, 14
8.6 # 1, 2, 3, 4, 5, 6, 8, 9
WEEK 9 (beginning Mar.07) Lecture: Direct Sums,
Cayley-Hamilton
Th..
8.6 # 10(b), 10(c), 10(d), 11, 12, 13, 14, 15, 17, 18, 19, 20,
22, 23, 24,
25(a), 25(b)
S App.A - all problems
WEEK 10 (beginning Mar.14, QUIZ # 2, covers 8.3, 8.4,
8.5)
Lecture:
Inner Product Spaces
9.1 # 1, 3, 4, 5, 6, 7, 9, 10, 12, 13, 14, 16, 18, 23, 25, 26,
27, 28, 30, 32
9.2 # 1, 2(b), 3(b), 4, 5, 6(c)-(f), 7, 8, 9, 10, 11, 13, 14,
15
WEEK 11 (beginning Mar.21) Lecture:
Spectral
Theorem
9.3 # 1, 2, 3, 4, 5(b), 5(d), 6, 8, 12, 13, 15
WEEK 12 (beginning Mar.28, QUIZ # 3, covers 8.6, 9.1)
Lecture:
Isometries
Unitary Spaces
9.4 # 1(c), 2(d), 2(f), 3(b), 3(d), 3(f), 6, 8, 10
S App.B # 1, 2, 3
(see problems in section "Unitary Spaces" in the Study Manual)
WEEK 13 (Apr.04) Lecture: Operators on
Unitary
Spaces
S App.B - all problems