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• Tentative syllabus: NOTE:The syllabus linked here is now out of date, For a discussion of the changes, see the Announcements section below.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 Cayley-Hamilton Theorem (proof not required), inner product spaces, orthogonal sets and orthonormal bases, Gram-Schmidt, orthogonal complements, adjoints of operators, self-adjoint 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 self-adjoint operator is diagonalizable by an orthonormal basis, and a key lemma for this result, that if T is self-adjoint then the orthogonal complement of a T-invariant subspace is T-invariant. (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 self-adjoint 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 write-up 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 VS1-VS8.
  • 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 10:30-11:30 AM and 1:40-3:00 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, 4.1-4.4, 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.