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
The GSI for this course is Rena Levitt. Her office hours
are:
Announcements:
Course Timetable (subject to change) |
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Date |
Topics |
Reading
|
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. |
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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 |
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M 5/12 |
Isometries of R^3 |
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W 5/14 |
Exterior Powers of a
vector space. Definition of determinant. |
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F 5/16 |
Properties of Determinants. |
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M 5/19 |
Characteristic Polynomial.
Multiplicity of Eigenvalues. |
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W 5/21 |
Nilpotent Operators. |
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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. |
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Wed 6/11 |
Take Home Final
Exam - Due 12:00 pm. |
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