Math 108A 
Intro to Linear Algebra  Spring 2009
Professor: Alex Dugas my
homepage
Office: 6510 South Hall
Office Hours: W 2  3, Th 10  11, F 11  12
Prerequisites: Math 5A, 8 (with a grade of C or better).
Texts: Sheldon Axler, Linear Algebra Done Right. Second
Edition. Springer
1997.
Other linear algebra texts are available for free online, and many of
these follow a more concrete matrix oriented approach, which will
probably look more familiar. Two options are:
 Sergei
Treil, Linear Algebra Done Wrong.
 Jim Hefferon, Linear Algebra.
Lecture: MWF
10:00
 10:50 pm. in Girv 2116.
The TA for this course is Jonathan Cass. His office is South Hall
6431U. His office hours are:
 M 11:30  12:30 pm
 T 5:00  7:00 pm (in MathLab 1607 SH)
Announcements:
 In case you missed
it, the final exam was a doozy. Here are my (unchecked) Solutions. Have a good summer!
 Extra Office Hours: Friday 6/5: 12
 1 in 6432P (Drew, subbing for Jon). I will hold some Friday 3 
4 and Sunday 11  1.
 The Final Exam is
next Monday 6/8: 8  11 am.
It will be cumulative, but will focus
slightly more on Chapters 3 and 5 of LADR. Here is a Practice Final that for you to work on. It
is roughly the same length and difficulty as the actual test. Solutions. 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, Kernel / Null Space, Image / Range,
Injective / OnetoOne, Surjective / Onto, Bijective / Invertible,
Isomorphic Vector Spaces, (Standard) Matrix of a
Linear Map, Coordinate Vector, Change of Basis Matrix,
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, 3.21; 5.6,
5.10,
5.13, 5.20, 5.21, 10.3.
 Of course, this
does not mean these are the only topics that may appear on the
test. But these are definitely the most important.
 Quiz 4 will be
Monday 6/1! Study the definitions of Eigenvalues,
Eigenvectors, Eigenspaces and Diagonalization.
 Quiz 3 will be
Wednesday 5/20! Study the definitions of the Standard Matrix of
a Linear Map: Mat(T, E_n,E_m); The Matrix of a Linear Map relative to
other bases: Mat(T; B_V, B_W); Coordinates of a vector in a basis B;
and the definition of a Change of Basis Matrix: Mat(I, B_new, B_old).
 Homework 6 (also posted below) has been
revised (Th 5/14 10:40am). Please be sure that you have the
current version.
 Midterm Solutions. Graded Exams will
be returned in class next week. The next homework assignment will
be posted this weekend (hopefully).
 The Midterm Exam
will be in class Friday 5/8. It will cover through p. 47, and
some of the section on Invertibility p. 5358. Here is a review sheet with some practice
problems. Solutions to the
review problems.
 Quiz #2 will be in
class Wednesday 4/29. It will cover Basis, Dimension, and Linear
Transformations.
 Solutions to
Homework 1 and Quiz 1 are now posted in the table below.
 The first quiz will
be Friday 4/17 at the start of class. It will be short (1015
min.) and you should study the definitions of Vector Space, Subspace,
Span, Sum of Subspaces and Linear (In)Dependence, and be able to
recognize when a given set or subset of vectors satisfies the
definitions.
 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

Homework


M 3/30

Introduction


Homework #1
Solutions


W 4/1

Complex Numbers

p. 23


F 4/3

Vector Spaces.
Definition and Axioms. Examples.

p. 410


M 4/6

Properties of Vector
Spaces. Subspaces.

p. 11  12


W 4/8

Subspaces.

p. 13  14

Homework #2
Solutions


F 4/10

Span of a set of vectors.
Sums of Subspaces

p. 14  15, 22


M 4/13

Direct Sums of Subspaces

p. 1518


W 4/15

Intro to Dimension.
Linear Independence.

p. 22  24

Homework #3
Solutions


F 4/17

QUIZ #1: Linear (In)Dependence, Span,
Subspaces
Sums. Solutions
Basis of a vector space.

p. 25  27


M 4/20

Basis and Dimension (cont.)

p. 27  31


W 4/22

Properties of Bases and
Dimension. Dimensions of Subspaces and Sums.

p. 31  34

Homework #4
Solutions


F 4/24

Practice Problems: Finding
Bases and Dimension.
Handout



M 4/27

Linear Transformations.

p. 37  41


W 4/ 29

QUIZ #2: Basis, Dimension, Linear
Transformations.
Solutions.
Solving Systems of Linear
Equations. Null Space/Kernel of a Linear Map.

p. 41  43

Homework 5
p. 5960: Ex. 4,
5, 7, 9, 10
Solutions


F 5/1

Image of a Linear
Map. Injectivity, Surjectivity.
Isomorphism of Vector Spaces. 
p. 43  44
p. 53  55
(Invertibility)


M 5/4

RankNullity Theorem.
Isomorphism and Dimension.

p. 45  47
p. 55, 57 (Theorems 3.18, 3.21)


W 5/6

Practice / Review




F
5/8

Midterm




M 5/11

Matrices. Standard
matrix of a linear map.

p. 48  50

Homework 6
Due: 5/20.


W 5/13

Coordinates. Matrix
of a linear map with respect to different bases.

p. 51  53


F 5/15

Change of Basis Matrices.

p. 214  216


M 5/18

Composition=Matrix
multiplication.
Change of Basis
Formula. 


W 5/20

Quiz 3: Matrix of a linear Map:
Standard matrix, change of basis matrix. Coordinates of a vector
relative to a basis.
Solutions.


Homework 7
Solutions


F 5/22

Eigenvalues, Eigenvectors,
Eigenspaces (Review)

Ch. 5
p. 7578


M 5/25

Memorial Day: No class!



W 5/27

Diagonalizaion.

p. 87  90
p. 79

Homework 8
Solutions


F 5/29

Thm 5.6 (Lemma from
Wednesday)
Applications of Diagonalization: Powers and Functions of
Matrices. Fibonacci Numbers.



M 6/1

Quiz 4: Eigenvectors,
Eigenvalues, Eigenspaces
Solutions
Nondiagonalizable
matrices. Existence of Complex Eigenvalues. Upper
Triangular Matrices.

p. 80  84


W 6/3

Upper Triangular Matrices.

p. 85  87


F 6/5

review




M 6/8

Final Exam  8:00
 11:00 am




Course Content and Goals:
We will cover Chapters 1,2,3 and 5 in the text: vector spaces, bases,
linear transformations, eigenvalues and eigenvectors. While you
have already studied these concepts in Math 5A, in this course we will
take a more abstract viewpoint and focus a great deal more on theory
and proofs. In terms of proof writing, this course is an
extension of Math 8: the homework and exams are meant to serve as an
opportunity for 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 (subject
to change). You may work together on homework problems; however,
you must write up your answers individually. You must justify
your answers 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.
Quizzes: 3 or 4 short
quizzes will be given in lecture over the course of the quarter.
Each will be about 10 to 15 minutes long and focus on definitions and
examples of key concepts. No makeup quizzes will be given, but
your lowest score will count only as extra credit. The dates of
the quizzes will be announced in advance and posted on the course
webpage.
Exams: There will be one
inclass midterm exam on
Friday May
8, 10:0010:50 am. Please arrive promptly. The
final exam is scheduled for
Monday
June 8, 8:00  11:00 am. No makeup 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 = 30%, Quizzes = 10%, Midterm =
20%, Final = 40%. 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.