Math 108A -
Intro to Linear Algebra - Spring 2008
Professor: Alex Dugas my
homepage
Office: 6510 South Hall
Office Hours: M 11 - 12, T 2:30 - 4
Prerequisites: Math 5A, 8 (with a grade of C or better).
Texts: Sheldon Axler, Linear Algebra Done Right. Springer
1997.
Alternative Text (Recommended): Sergei Treil, Linear Algebra Done Wrong.
Lecture: MWF 12:00
- 12:50 pm. in Arts 1251.
The GSI for this course is Charles Martin. His office hours
are:
- MW 1 - 2 pm , 6432F
- Th 5 - 7 pm ,
Mathlab 1607 SH
Announcements:
- My Office Hours next
week will be Monday 2:30 - 4:30.
- The Final Exam is
next Tuesday 6/10 12-3pm. It will be cumulative, but will focus
slightly more on Chapter 5 of LADR. Here is an old exam that you may practice. The most
important things for you to review are the Definitions and Major
Theorems:
- Defintions:
Subspace, Sum, Direct Sum, Span, Linear (In)Dependence, Basis,
Dimension, Linear Map, Null Space, Range, Invertible, Matrix of a
Linear Map, Eigenvalue, Eigenvector, Eigenspace, Diagonalizable.
- Major Theorems:
2.10, 2.11, 2.12, 2.14, 2.15, 2.16, 2.17; 3.2, 3.4, 3.18; 5.6, 5.10,
5.13, 5.21(a,b).
- Of course, this
does not mean these are the only topics that may appear on the
test. But these are definitely the most important.
- Solutions to Midterm 2.
- Here is Midterm 2. It is due next Friday, May 23,
by 4pm.
- If you want to
improve your midterm 1 score, do this optional
homework. Please read it carefully and follow all
instructions. It is due by Friday 5/2. To get your midterm
back, contact the TA Charles Martin.
- Midterm 1 will cover
chapters 1 and 2 of LADR.
- The most important
topics include
- Vector spaces (know the definition and how to tell if
something is a vector space or not.)
- Subspaces (Be able to prove that a subset is or is not a
subspace.)
- Sums and Direct Sums of Subspaces (Know the definitions.
Be able to compute the sum of two or more subspaces, and determine
whether the sum is direct.)
- Span of a set of vectors.
- Linear (In)Dependence of vectors.
- Bases for vector spaces (Know the definition and how to
find a basis. Be able to show that certain vectors form a basis.)
- Dimension of vector spaces (Know how to compute the dimension
of a vector space, and understand how this relates to bases, spans and
linearly independent sets of vectors.)
- As a practice Exam, do exercises 2.2, 2.3 and 2.6 on p. 10 of
LADW and exercise 8 on p. 19 of LADR. Practice
Exam Solutions
- Homeworks and
readings are displayed in the table below. "R" refers to the text
"Linear algebra done RIGHT", and "W" refers to the text "Linear algebra
done WRONG" (see the link above). Please note that page numbers
in W refer to the numbering of the text, and NOT of the pdf file.
- Each of
you should come to my office hours at least once during the first three
weeks of class to introduce yourself. If the scheduled times do
not fit your schedule, please email me or talk to me after class to
arrange another time.
|
|
Date
|
Topics
|
Reading
R = LADR
W = LADW
(-) = Optional
|
Homework
|
|
M 3/31
|
Intro / Review
|
|
Homework 1
|
|
W 4/2
|
Complex Numbers
|
R: p. 2-3
|
|
F 4/4
|
Vector Spaces
|
R: p. 4-10
|
|
M 4/7
|
Properties of Vector
Spaces. Subspaces.
|
R: p. 11 - 14
|
Homework 2
|
|
W 4/9
|
Sums of Subspaces.
Span.
|
R: p. 14
|
|
F 4/11
|
Direct Sums.
|
R: p. 15 - 18
|
|
M 4/14
|
Linear Independence.
|
R: p. 22 - 25
|
|
W 4/16
|
Dimension.
|
R: p. 25 - 27
|
Homework 3
|
|
F 4/18
|
Bases.
|
R: p. 28 - 32
|
|
M 4/21
|
Equivalent Characterizations of Bases.
|
|
|
W 4/23
|
Dimensions of sums of
subspaces
|
R: p. 33-34
|
|
F
4/25
|
Midterm
1
|
|
|
|
M 4/28
|
Linear Maps.
|
R: p. 38-41
|
Homework 4:
R: p. 59-60:
Ex. 2, 4, 5, 7, 9, 12
|
|
W 4/30
|
Null Space of a linear
map.
|
R: p. 41-43
|
|
F 5/2
|
Range of a linear map.
|
R: p. 43-44
|
|
M 5/5
|
Rank-Nullity Theorem
|
R: p. 45
|
|
W 5/7
|
Systems of linear
equations.
|
R: p. 46-47
|
Homework 5
|
|
F 5/9
|
Invertibility.
Isomorphisms.
|
R: p. 54-55
|
|
M 5/12
|
Matrices associated to
linear maps.
|
R: p. 48-50
|
|
W 5/14
|
Matrices and linear maps
(cont.)
|
|
|
|
F 5/16
|
Composition of linear maps
and matrix
multiplication. Invertible matrices.
|
R: p. 51-53
|
|
|
M 5/19
|
Change of Bases.
|
R: p. 214-216
|
|
|
W 5/21
|
Eigenvalues, Eigenvectors,
Eigenspaces
|
R: p. 76-79
|
|
|
F
5/23
|
Invariant Subspaces.
Upper-Triangular Matrices
|
R: p. 82-84
|
|
|
M 5/26
|
Memorial Day Holiday: No Class
|
|
Homework 6
LADR: p. 94-5:
5, 7, 8, 9, 10, 11
|
W 5/28
|
Existence of eigenvalues
over C. Upper-triangular form of an operator.
|
R : p. 81, 84 - 85
|
|
|
F 5/30
|
Eigenvalues of
upper-triangular matrices.
|
R: p. 85 - 87
|
|
|
M 6/2
|
Diagonal matrices.
Diagonalization.
|
R: p. 87 - 90
|
|
|
W 6/4
|
Invariant subspaces of
R-vector spaces
|
R: p. 91 - 93
|
|
|
F 6/6
|
Spectral theory using
Determinants (Summary.)
|
|
|
|
Tu 6/10
|
Final Exam - 12:00
- 3:00 pm
|
|
|
|
Course Content and Goals: We will cover Chapters 1,2,3 and 5
in LADR (vector spaces, linear transformations, eigenvalues and
eigenvectors) and additional topics from LADW (inverse matrices,
determinants, change of basis) as time allows. LADR emphasizes
abstract properties of finite dimensional vector spaces and linear
transformations, while LADW presents a more concrete matrix oriented
approach. I will try to combine these two perspectives, often
asking you to read the texts side by side to compare. In terms of
proof writing, this course serves as an extension of Math 8. On
the homeworks and exams, I expect you to further practice and develop
your logical reasoning and proof writing skills.
Homework: Homework exercises will be assigned in lecture
and listed on the course webpage. Homework will be due in lecture
each Wednesday. You may work together on homework problems;
however, you must write up your answers individually. You must
show all your work and clearly explain your reasoning in order to
receive full credit.
Late
homeworks will not be accepted.
However,
your lowest homework score will be automatically dropped.
Exams: There will be two in-class midterm exam on
Friday April 25, 12:00--12:50 pm and
Wednesday May 21, 12:00--12:50 pm.
Please arrive promptly. The final exam is scheduled for
Tuesday June 10, 12:00--3:00 pm.
The second midterm and/or final exam may be take-home (to be decided
later). The problems on the exams will be similar to those seen
in class or on the homeworks. No make-up exams will be
given, except in extraordinary circumstances. If you have a
serious conflict with any of these exams or miss one for any reason, it
is your responsibility to notify me immediately so that other
arrangements may be made
Grades: Grades will be computed from your scores on
homeworks
and exams as
follows:
Homework
= 25%,
Midterms = 20% each, Final = 35%. No letter grades will be
assigned
until the
end of the semester, and the exact grading scale will be curved
relative to the
difficulty
of the exams. However,
a 90% or
above will
guarantee you at least an A, an 80% will be at least a B, and 70% will
be at
least a C.