Math 110: Linear Algebra
UC Berkeley, Dept. of Mathematics, Spring 2013
Syllabus
Instructor: Benoit Dherin
Oce: 791 Evans Hall
Email: bdherin(at)berkeley(dot)edu
Course Homepage: math.berkeley.edu/dherin/Math110_2013.html or on bspace
1 Course Info
Starts: Wednesday, January 23, 2013
Location: 100 Lewis
Lectures: MWF 23 p.m.
Oce Hours: Mondays 810 a.m. or by appointment
Prerequisites: Math 54 or a course with equivalent linear algebra content
Textbook: Linear Algebra Done Right, by Sheldon Axler (Springer, 2nd ed.)
CCN: 54151
Discussion Sections: Wednesdays, see Times and Places
Grading Policies: 20% homework, 20% each midterm, 40% nal exam. If you
miss one for any reason, it is eectively a 0 grade. You are allowed to
replace one of the midterm grades with your nal exam grade, but there are
no makeup midterm exams.
Course Policies: There will be no makeup midterms or nal exams. No late
homework will be accepted. Grades of Incomplete will be granted only for
dire medical or personal emergencies that cause you to miss the nal, and
only if your work up to that point has been satisfactory.
Academic Honesty: You are expected to rely on your own knowledge and abil-
ity and not to use unauthorized materials or represent the work of others as
your own.
1
2 Exams
Midterm 1: Wednesday, February 20, 2013
Midterm 2: Wednesday, March 20, 2013
Final Exam: Tuesday, May 14, 2013
3 Homework
Homework assignments will be posted on this site and on bspace each Wednesday.
They should be turned in to your GSI by the following Wednesday at the beginning
of your discussion section. Each assignment will consist of a reading assignment
and a series of exercises related to the reading. The exercises will at times be
slightly ahead of the lecture in order to encourage self-study as well as discussions
on Piazza, where you are encouraged to post and answer questions.
4 Description
We will closely follow the textbook
Linear Algebra Done Right, by Sheldon Axler (Springer 2nd ed.)
Here is an outline of the topics we will encounter:
Vector Spaces: Denition, Properties, Subspaces, Sums and Direct Sums
Finite-Dimensional Vector Spaces: Span and Linear Independence,
Base, Dimension
Linear Maps: Denitions, Examples, Null Spaces and Ranges, Matrices,
Invertibility
Eigenvalues and Eigenvectors: Denitions, Invariant Subspaces, Upper-
Triangular & Diagonal Matrices
Inner-Product Spaces: Inner Products, Orthogonal Bases and Projec-
tors, Linear Functional and Adjoints
Operators on Inner-Product Spaces: Self-Adjoint, Normal, & Positive
Operators, Spectral Theorem, Isometries, Polar & Singular-Value Decom-
position
Operators on Complex Vector Spaces: Generalized Eigenvectors, Char-
acteristic Polynomial, Decomposition of an Operator, Square Roots, Mini-
mal Polynomial, Jordan Form
2
Operators on Real Vector Spaces: Eigenvalues of Square Matrices,
Block Upper-Triangular Matrices, Characteristic Polynomial, Jordan Form
Trace and Determinant: Change of Basis, Trace, Determinant of an
Operator, Determinant of a Matrix, Volume
Math 110 is ideal as an upper-division entrance course: the basic concepts (vector
spaces, linear maps, matrices, eigenvectors and eigenvalues, etc.) will have already
been studied in a lower-division course (e.g., Math 54), but in a very concrete way
and with an emphasis on computation. In Math 110, the focus is more on abstract
denitions and formal proofs, with an emphasis on classication problems (which
are ubiquitous in mathematics) and normal forms. The courses culminating point
will then be the Jordan normal form theorem, which classies and gives a normal
form for operators on a complex vector space.
If time permits, we will make some excursions to more theoretical topics (ab-
stract algebra, category theory, set theory, etc.) to familiarize students with the
abstract way of doing mathematics, and possibly visit some more concrete topics
(quantum mechanics, quantum computing, machine learning, PageRank, etc.) in
order to get a sense of the sheer power of linear algebra in real-life applications.
These excursions will be clearly indicated to the audience at the beginning of the
lecture. They will not be part of the homework nor of the exam material. Their
only purpose will be to (hopefully!) foster enthusiasm.
A good additional resource is the article
Down with Determinants, by Sheldon Axler,
which summarizes the textbook content in only 18 pages. It can be found at
http://www.axler.net/DwD.html
5 Summary
Lecture 1 (01/23/2013): Introduction to the class: textbook, sections,
grading, homework, Bspace, Piazza, etc.
3
