Linear Algebra
18.700 Fall Semester, 2014
General Information
Class meetings: Tuesday and Thursday 9:30–11:00, in 4-163.
Text: Sheldon Axler, Linear Algebra Done Right. Read the text before class as well as after;
your understanding and your chance of catching me in a faux pas will both be greatly increased.
Lecturer: David Vogan, E17-442. Telephone: 617-253-4991. E-mail: dav@math.mit.edu. My
office hours are Wednesday 3–4, Thursday 11:30–1:00, or by appointment.
Homework: There will be nine graded problem sets; due dates IN CLASS are on the
schedule below. Late problem sets will not be accepted. (Really. This is partly a logistical issue
about getting the problem sets to the grader.)
Exams: There will be three eighty-minute exams during the lecture hour: Sept 25, Oct 21,
and Nov 18. There will be a three-hour final exam Wednesday, December 17, 9:00–12:00 in
4-163. All exams will be closed book.
Grading: Each hour exam will be worth 100 points, the final exam will be worth 200 points,
and the problem sets will be worth about 20 points each.
Readings: A few topics not fully covered in the text are discussed in supplementary notes
available on the class website math.mit.edu/∼dav/700.html. At the moment these include Notes
on finite fields (written F in the syllabus below); Notes on Gaussian elimination (GE); Notes on
orthogonal bases (OB); and Notes on generalized eigenvalues (EIG).
Schedule
Thur 9/4
Lecture 1
Ch 1, 2–12
Def of vector spaces, properties
Tues 9/9
Thur 9/11
Lecture 2
Lecture 3
Ch 1, 13–18
Ch 2, 21–31
Subspaces, sums and direct sums
Span, independence, bases
Tues 9/16
Thur 9/18
Lecture 4
Lecture 5
Ch 2, 31—34
Ch 3, 38–47
Bases and dimension
Linear maps, null space/range
PS 1 due
Tues 9/23
Thur 9/25
Lecture 6
Lecture 7
Ch 3, 48–58
matrices, invertibility
Exam 1 on Chapters 1–3
PS 2 due
Tues 9/30
Thur 10/2
Lecture 8
Lecture 9
F, GE
GE
Finite fields. Systems of equations.
Gaussian elimination
Tues 10/7
Thur 10/9
Lecture 10 F, Ch 5, 75–79
Lecture 11 Ch 5, 80–81
Counting over Fp . Invariant subspaces. PS 3 due
Finding eigenvectors
Tues 10/14
Lecture 12
Ch 5, 81–90
Upper triangular and diagonal matrices PS 4 due
Thur 10/16
Lecture 13
Ch 5, 91–93, F
Eigenvectors over R and Fp
Tues 10/21 Lecture 14
Thur 10/23 Lecture 15
Exam 2 on Chapters 1–5
Ch 6, 97–111, OB Inner products, Gram-Schmidt
Tues 10/28 Lecture 16 Ch 6, 111–116
Thur 10/30 Lecture 17 Ch 7, 127–132
Orthogonal projection, minimization
Adjoint, self-adjoint, normal
PS 5 due
Tues 11/4
Thur 11/6
Spectral theorem
Positive operators
PS 6 due
Lecture 18 Ch 7, 132–144
Lecture 19 Ch 7, 144–147
Tues 11/11 Holiday
Thur 11/13 Lecture 20
Ch 7, 147–157
Veterans Day
Isometries, polar decomposition
PS 7 due
Tues 11/18 Lecture 21
Thur 11/20 Lecture 22
Tues 11/25 Lecture 23
Thur 11/27 Holiday
Ch 8, 163–168
Ch 8, 173–178
EXAM 3 on Chapters 1–7
Generalized eigenspaces
Generalized eigenspace decomposition
Thanksgiving
PS 8 due
PS 9 due
Tues 12/2
Thur 12/4
Lecture 24 Ch 8, 168–173
Lecture 25 Ch 10,
Characteristic polynomial
Determinant
Tues 12/9
Lecture 26
Trace, canonical commutation relations
week of 12/15–12/19
Ch 10,
Final Exam