Syllabus
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The course will cover linear independence, rank, orthogonality and projections, orthonormal bases, the four fundamental subspaces of a matrix, least-squares, the QR decomposition, eigenvalues of symmetric matrices, the singular value decomposition, the condition number of a matrix, algorithms for solving linear systems.
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The course intends to provide the mathematical background in matrix
theory for modern methods of data analysis, scientific computing, and
other applications to science and technology. While the choice of
topics is geared towards integration with other disciplines, the
emphasis of the lecture and the homework assignments will be on
theory, like in other mathematics courses.
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Math 104 is an entirely new class that replaces Math 103. All rumors
concerning 103 are now obsolete :) You can understand "matrix theory"
as being the same as "linear algebra", but Math 104 will mostly differ
from the other class Math 113 in the choice of topics: very roughly
speaking, Math 104 will be to "analysis" what Math 113 is to
"algebra".
Textbook
The required textbook is Numerical Linear Algebra by
L. N. Trefethen and D. Bau III, ed. SIAM (1997). It will NOT be
available at the bookstore. Instead, the plan is to order it
online with a decent discount from the editor. I recommend going
through the following steps:
Become a student member of the Society for Industrial and Applied
Mathematics (SIAM) by signing up here.
It's free and as a perk you will receive the 8-page
newsjournal "SIAM news" once every 2 months.
With student membership you have a 30 percent discount on all
SIAM books, including the class textbook. Order the book here and
don't forget to scroll down choose "SIAM member price" over "List
price" at the bottom of the page. Make sure you specify "Member price
Math 104 Stanford" in the Special Instruction box before submitting
your order. If one of the steps of ordering online poses an
insurmountable difficulty to you, contact the teaching staff for help.
If you wish to purchase the book elsewhere, or if you wish to group
orders among yourselves, that is fine. One copy of the book will be on
reserve at the Math/CS library.
The lecture may be inspired by material from other books, but typeset
notes will be provided everytime this happens. One of these books is
Introduction to Linear Algebra by G. Strang. It's a great book,
but consider it an optional purchase in the scope of the course.
Link to the class notes: December 1
draft. Older versions: 10/14, 10/06, 09/24.
Link to the first few lectures in Trefethen's book: here.
Prerequisites
Math 51, and either Math 52 or Math 53. Alternatively, familiarity with the following notions:
Vector operations: dot product, cross product;
Matrix operations: matrix-matrix multiplication, matrix-vector multiplication;
Partial derivatives and the chain rule of vector calculus;
Definition of eigenvalue and eigenvector;
3-by-3 determinants.
No knowledge of computer programming is necessary. There will be no programming assignment.
