Math 115AH Linear Algebra Lect. : MWF 2:00-2:50 Disc. Tue.: 2:00-2:50 in MS 5117 |
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Instructor: Olga Radko (radko@math.ucla.edu) Office hours: M 3-3:30; W 3-5; F 11:30-12 in MS 5366 |
Teaching assistant: Julia Dobrosotsky (juliadobro@yahoo.com) Office hours: T 2-3; R 12-1 in Math Center; R 4-5. |
Course Information.
COURSE OUTLINE: This course is an
honors introductory course in abstract Linear Algebra. The goal of
the course is to study vector spaces, linear transformations and
inner product spaces, finishing with the Spectral Theorem. The
emphasis of the class is on learning (on the material of linear
algebra) how to understand mathematical concepts and how to prove
rigorous mathematical statements.
TEXTBOOK:
Paul R. Halmos, ``Finite-Dimensional Vector spaces'', Springer,
Undergraduate Texts in mathematics. It is a good idea to read the
textbook concurrently with the course, and perhaps a few pages ahead
when possible.
OTHER BOOKS YOU MIGHT FIND
USEFUL:
1. K. Hoffman, R. Kunze, ``Linear algebra'', Prentice Hall, second ed.
2. S. Lipschutz ``Schaum's outline of theory and problems of linear algebra''.
3. D. Solow ``How to read and do proofs: an introduction to mathematical thought processes''.
4. G. Polya `` Mathematical discovery: on understanding, learning and teaching problem solving''. See also other books by this author.
5. A. Cupillari `` The nuts and bolts of proofs''.
Some of these books are put on the reserve at the libraury.
HOMEWORK: Homework
is an essential part of the course, since trying to solve a lot of
different problems on your own is the only way to learn how to come
up with proofs and write them down. Homework will be assigned weekly
and collected on Friday in class. In addition to problems from the
book, there will be some problems given to you in class (and
available on the class web page). Only three of the problems, chosen
at random, will be graded each week. However, it is recommended to
try to solve each of the assigned problems. Two lowest HW scores will
be dropped.
EXAMS AND QUIZZES: Midterm:
November 5th (Wednesday), in class; Final: December, 9th. There will
be no make-up exams. Throughout the semester, there will be several
quizzes in the discussion section. Quiz dates and topics will be
announced in advance. The lowest quiz score will be dropped.
GRADING: Your grade will be
computed as the best of the following:
HW (20%) + Midterm(20%)+Quizzes(20%)+Final (40%);
HW (20%) + Midterm(15%)+ Quizzes(15%) + Final (50%);
Class Handouts.
Homework assignments.
Lectures |
Material |
Sections |
Homework |
Due Date |
1 |
Fields, Vector Spaces |
1-4 |
p. 2: 1,2,3; p.6: 1,2; p. 12: 1,2,3,5,8,9; Additional problems |
10/3 |
2 |
Subspaces |
10 |
||
3 |
Spans and Linear Combinations |
5-6 |
||
4 |
Bases and Dimension |
8, 11, 12 |
p. 16: 1,3; p 18: 3-8; p. 22: 1,3,5,6; Additional problems |
10/10 |
5 |
Isomorphisms |
9 |
||
6 |
Dual Spaces |
13-15 |
||
7 |
Annihilators and Reflexivity |
16, 17 |
pp. 27-28: 2,4,6; p.32: 1,3,5; p. 57: 1ab; Additional problems |
10/17 |
8 |
Direct Sums |
18-20 |
||
9 |
Linear Transformations |
32, 33 |
||
10 |
Algebra of Linear Transformations |
34-36 |
p. 57: 4,5; p. 61: 1,3,7; pp. 63-64: 2-10; pp. 68-70: 1,3,4,7,11; p. 73: 1-3; Additional problems |
10/24 |
11 |
Matrices of Linear Transformations |
37, 38 |
||
12 |
Reduction of a Linear Transformation |
39, 40 |
||
13 |
Projections and Invariance |
41, 43 |
Required: p. 77:
2,3,5,6,7,9; p. 86: 1, 2, 4; p. 90: 1, 2; p. 94: 1, 4, 5 and Additonal problems; |
10/31 |
14 |
Adjoints, Change of Basis, Similarity |
44-47 |
||
15 |
Range and Null Space |
49-51 |
||
16 |
Eigenvalues and Eigenvectors |
54-55 |
p. 106: 1abd,3,5,7; p.108: 1adf,2; Additional Problems |
11/7 |
17 |
Triangular Form of a Transformation |
56 |
||
18 |
Midterm on Wed. 11/05 |
Covers lectures 1-15 |
||
19 |
Inner Product Spaces |
59-62 |
pp. 123-124: 2-4; pp. 128-129: 1-3,5,6a; p. 134: 1,3,4a; Additoinal Problems |
11/14 |
20 |
Completeness, Schwarz' Inequality |
63-65 |
||
21 |
Projections and Self-duality |
66-69 |
||
22 |
Self-adjoint Linear Transformations |
70 |
p. 137: 1,2,3,6,9,10; pp. 141-142: 1,2abd,3,5,7; p. 145: 2,3,5-7; Additional Problems |
11/21 |
23 |
Polarization, Positivity |
71, 72 |
||
24 |
Orthonormal Bases, Gramm-Schmidt |
73, 74 |
||
25 |
Orthogonal Projections |
75 |
pp. 149-150: 1-4; p. 153: 1,3,6; p. 155: 1,2; p. 158: 1,2,4; Additional Problems |
11/28 |
26 |
Complexification, Spectrum |
77, 78 |
||
27 |
Spectral Theorem |
79 |
||
28 |
Normal and Orthogonal Transformations |
80,81,... |
TBA |
12/05 |
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