640:350:02
Linear Algebra
FALL 2016
Basic information
 Instructor: Professor Eugene Speer
 Hill Center 520
 848–445–7974
 speer at math.rutgers.edu
 Office hours:
 Monday, 10:30–11:30 AM, Hill 520
 Wednesday, 1:40–3:00 PM, Hill 520
 Thursday, 10:30–11:30 AM, Hill 520
 Or by appointment or chance in Hill 520
 Detailed course policies and information:
As web page and as a PDF file.
 Tentative syllabus: NOTE:The syllabus
linked here is now out of date, For a discussion of the changes,
see the Announcements section below.
Here are the original links to the tentative syllabus:
as a web page and as
a PDF file.
 Hints on writing proofs:
 From your instructor: As web page and as
a PDF file.
 From Professor James Munkres, the author
of the text used in Math 441, Introductory Topology. (The copy here was
obtained from Prof. Jim Wiseman at Agnes Scott College; I do not know who
wrote the prefatory remarks.)
 Homework assignments:
Click here for assignments.
Final Exam
 The final exam will be held on Friday, December 23, 12:00–3:00
PM, in SEC 206.
 The exam will cover all our work in the course, possibly with an
emphasis on material covered since the second midterm.
 We will hold a review/question session on Thursday, December 15,
1:30–3:00 PM, in Hill 425. Come prepared to ask questions' I
will not present a systematic review.
 I will not hold my usual office hours once classes are over.
However, I will hold special office hours on Thursday, 12/22,
10:00–11:00 AM and 2:00–3:00 PM, in Hill 520.
 Here are a few miscellaneous comments:
 You should know the same definitions and results which were
specified for the first two exams (still posted below).
 You will be given the formulas for changes of coordinates which were
given on the second midterm.
 You should know the definitions of, and basic facts about: invariant
subspaces, the CayleyHamilton Theorem (proof not required), inner
product spaces, orthogonal sets and orthonormal bases, GramSchmidt,
orthogonal complements, adjoints of operators, selfadjoint and normal
operators and their diagonalizability, and unitary and orthogonal
operators and matrices.
 You should be able to give proofs of simple results about all the
material in the course. For more complicated proofs: in addition to
those mentioned for previous exams you should be able to prove that a
selfadjoint operator is diagonalizable by an orthonormal basis, and a key
lemma for this result, that if T is selfadjoint then the orthogonal
complement of a Tinvariant subspace is Tinvariant. (You are not
responsible for the proof that a normal operator is diagonalizable.)
 You should know the basic facts about sums and direct sums of subspaces,
covered on Assignment 6 and used later in the course.
 This list is not complete; there are various minor results not
included which you should know or be able to reconstruct.
Announcements and additional resources
 12/14/2016: Here notes on the approach we took in class to the
diagonalizability of selfadjoint and normal
operators.
 12/13/2016: We did not move as fast at the end of the course
as I had expected (see the announcement of 11/21 below). In the end we
covered Sections 6.1 through 6.5 and did nothing from Chapter 7; we may
discuss material from Section 6.6 on Wednesday, 12/14, but it will not be
on the final.
 11/21/2016: At the moment we are one lecture behind the pace
to which we aspired in the original syllabus. As a result, we will
change the order in which we cover the last two topics of the course.
We will begin today to discuss inner product spaces (Chapter 6),
covering Sections 6.1–6.6 in about five lectures. We will
then spend the remaining time on the Jordan Canonical Form, Chapter 7.
 11/01/2016: Here is a writeup of
the proof of Theorem 4.4 which was given in
class on 10/31.
 9/07/2016: Here are the review notes on
Math 250 discussed in the material on course policies and information
(see the links above).
Exam 1
 The first midterm exam will be held on Wednesday, October 12.
 The exam will cover our work through Section 2.4 of the text. Almost
all of this was covered by Wednesday, October 5, but we will finish a bit
of it on Monday, October 10.
 Here are a few miscellaneous comments:

You should know the major definitions: subspace, span, linear
independence and dependence, basis, linear transformation. You should
know and be able to work with such concepts and in particular does not
include as the null space and range of
a linear transformation, the coordinate representation of vectors and
linear transformations with respect to an ordered basis or bases, the
inverse of a linear transformation, the operator L_A associated with a
matrix A, etc.
You do not need to know the formal definition of a vector space, that is,
you do not need to memorize the eight axioms VS1VS8.
 You should be familiar with, and be able to use, important results
such as Theorems 1.3, Theorem 1.8*, Theorem 1.9, Corollaries 1 and 2 of
Theorem 1.10, Theorem 2.3*, Theorem 2.4*, Theorem 2.6, Theorem 2.15, and
Theorem 2.19*. For the theorems in this list which are starred you
should be able to give proofs. This list is not complete; there are
various minor results not included which you should know or be able to
reconstruct.
 I will hold extra office hours on Wednesday, October 12; my
office hours that day will be 10:3011:30 AM and 1:403:00 PM.
Exam 2
 The second midterm exam will be held on Wednesday, November 16.
 The exam will cover our work since the first exam, that is, Sections
2.5, 3.1–3.4, 4.14.4, 5.1, and 5.2. (Note: Sections 3.3 and
3.4 were added to this list on November 14.) However, there is of course
much from the first part of the course that will be needed to answer
questions about this later material.
 Here are a few miscellaneous comments:

You should know about change of coordinates. You will be given the formula
for the change of coordinate matrix Q and for its action on coordinate
vectors; see for example the formulas in Theorem 2.22.
 You should know the definitions of the determinant and of eigenvalues,
eigenvectors, eigenspaces, and algebraic and geometric multiplicities.
 You should know and understand the main theorem about
diagonalizability proved in class on November 9 (this is included in the
solutions to Assignment 10, and is equivalent to the "Test for
Diagonalization" on page 269 of the text), and the key ideas in its proof
(for example, you should be able to prove the closely related Theorems
5.5 and 5.8).
This list is not complete; there are various minor results not
included which you should know or be able to reconstruct.
 I will hold additional office hours for this class on Wednesday,
November 16, 10:00–11:00 AM, as well as my usual office hours
Wednesday afternoon.