Math 108B - Advanced Linear Algebra - Spring 2008

Professor: Alex Dugas my homepage
Office: 6510 South Hall
Office Hours: M 11 - 12, W 1:30 - 3

Prerequisites: Math 108A (with a grade of C or better).

Texts: Sheldon Axler, Linear Algebra Done Right. Springer 1997.
Alternative Text (Recommended): Sergei Treil, Linear Algebra Done Wrong.

Lecture: MWF 9:00 - 9:50 am. in Arts 1251.

The GSI for this course is Rena Levitt.  Her office hours are:

• M 2 - 3 pm , 6431D
•  T 5 - 7 pm , Mathlab  1607 SH

Announcements:

• Office Hours for Finals week:  Rena: Mon 12-3.   Alex: Tu 10-11, 3:30-4:30.
  
• Here is your Final Examination.  Please turn it in to me no later than 12pm Wednesday 6/11.
  
• Rena's office hours this week have been rescheduled.  They will be M (6/2) 12-2 and F (6/6) 10-11,  in her office.
  
• At long last, here are the Solutions to Midterm 2.
  
• Here is Midterm 2.  It is due in class next Friday, May 23rd.
  
• Midterm 1 Solutions.
  
• Homeworks and readings are displayed in the table below.  "R" refers to the text "Linear algebra done RIGHT", and "W" refers to the text "Linear algebra done WRONG" (see the link above).  Please note that page numbers in W refer to the numbering of the text, and NOT of the pdf file.
  
• Each of 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	Reading R = LADR W = LADW (-) = Optional	Homework
M 3/31	Intro / Review		Homework 1 Solutions
W 4/2	Matrix of a linear transformation.  Change of Bases.	W: p. 62-65
F 4/4	Change of Bases (cont.)  Similar Matrices.	W: p. 65-66
M 4/7	Inner Product Spaces.	R: p. 97-101
W 4/9	Norms.  Cauchy-Schwarz, Triangle  Inequalities.	R: p. 102 - 106  (W: p. 109-115)	Homework 2 Solutions
F 4/11	Parallelogram Identity.  Normed Spaces.   Orthonormal Bases	R:  p.  106 - 108
M 4/14	Gram-Schmidt Process.	R: p. 108 - 110
W 4/16	Orthogonal Projection.	R: p.  111 - 116 (W : p. 117-124)	Homework 3 Solutions
F 4/18	Minimization via Orthog. Proj.
M 4/21	Linear Functionals.  Dual Spaces.	R: p. 117-118	Homework 4: R: p. 125,    Ex. 24, 27, 29 R: p. 158,     Ex. 1, 4, 8, 9 Solutions
W 4/23	Adjoints	R: p. 118-121
F 4/25	Self-Adjoint Operators.  Isometries.	R: p.  128
M  4/28	Matrices of Isometries.  Rotations and Reflections.	R: p. 147-49
W 4/30	Normal Operators	R: p.129-132
F 5/2	Complex Spectral Theorem	R: p. 132-134
M 5/5	Real Spectral Theorem	R: p. 134-136	Homework 5 Solutions
W 5/7	Real Spectral Theorem (cont.)	R: p. 136-137
F 5/9	Spectral Theorems for Matrices.  Isometries over C and R^2
M 5/12	Isometries of R^3
W 5/14	Exterior Powers of a vector space.  Definition of determinant.
F 5/16	Properties of Determinants.
M 5/19	Characteristic Polynomial. Multiplicity of Eigenvalues.
W 5/21	Nilpotent Operators.
F 5/23	Generalized Eigenvectors and Eigenspaces.	R: p. 164-167
M 5/26	Memorial Day Holiday: No Class		Homework 6 Solutions
W 5/28	V = null(T-xI)^n + range(T-xI)^n,  (direct sum)  where x= eigenvalue of T.	R: p. 167 (R : p. 168-173)
F 5/30	1) V is the direct sum of the generalized eigenspaces of T.  2) multiplicity of eigenvalue x = dimension of gen. eigenspace Vx.   (Proofs in lecture will differ from proofs in the book, since we defined multiplicity of eigenvalues differently)	R : p. 173-176 (L: p. 195-200)
M 6/2	Jordan Normal form.  Jordan Basis.  Reduction to nilpotent case.	R: p. 183, 186-7
W 6/4	Jordan Bases for nilpotent operators.	R: p. 183-186
F 6/6	Examples of Jordan normal form.  Cayley Hamilton Theorem.
Wed 6/11	Take Home Final Exam - Due 12:00 pm.

Course Content and Goals:  This course is a continuation of Math 108A, and we will focus on the two following problems.

• When is a matrix/linear transformation diagonalizable?
• If a linear transformation is not diagonalizable, what is the simplest possible form of its matrix (in some basis)?

We will cover most of Chapters 6, 7 and 8 in LADR, along with additional topics from LADW as time allows.  This includes inner product spaces, the spectral theorem, and Jordan normal form.  LADR emphasizes abstract properties of finite dimensional vector spaces and linear transformations, while LADW presents a more concrete matrix oriented approach.  I will try to combine these two perspectives, often asking you to read the texts side by side to compare.  Like 108A, the homework and exams in this course will tend to be theoretical and often proof-based.

Homework:  Homework exercises will be assigned in lecture and listed on the course webpage, to be submitted in lecture each week.  You may work together on homework problems; however, you must write up your answers individually.  You must show all your work 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.

Exams: There will be one or two take-home midterm exams throughout the quarter.  The final exam is scheduled for Wednesday June 11, 8:00--11:00 am.  The final exam may also be take-home (to be decided later).  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%, Midterm(s) = 30%, Final = 40%.  The exact grading scale will be curved; however,  an 85% or above will guarantee you at least an A, 70% will be at least a B, and 55% will be at least a C.