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:

1.  Sergei Treil, Linear Algebra Done Wrong.
2. 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 6431-U.   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 / One-to-One, 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. 53-58.  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 (10-15 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.

# Course Timetable (subject to change)

Date

Topics

Homework

M 3/30
Introduction

Homework #1
Solutions

W 4/1
Complex Numbers
p. 2-3

F 4/3
Vector Spaces.  Definition and Axioms.  Examples.
p. 4-10

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. 15-18

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. 59-60: 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
Rank-Nullity 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. 75-78

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
Non-diagonalizable 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 make-up 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 in-class midterm exam on Friday May 8, 10:00--10:50 am.  Please arrive promptly.  The final exam is scheduled for Monday June 8, 8:00 -- 11:00 am.  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 = 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.