Math 417: Matrix Algebra

Professor: David E Speyer, speyer@umich.edu Class: 539 Dennison, Monday, Wednesday, Friday 1-2 PM Office hours: 2844 East Hall, Monday 10-12, Tuesday 2-3, at other times by appointment Exams: Midterm 1: Friday October 4               Midterm 2: Monday November 4               Final Exam: December 13, 4-6 PM in Dennison 296 Problem Sets due: Wednesdays in class Textbook: Linear Algebra with Applications, Fifth edition, Otto Bretscher, ISBN 9780321796974. The fifth edition will be the standard reference for readings and homework; it is available at Ulrich's and many other bookstores. Since a significant proportion of the class has the fourth edition; I will try to give corresponding references in the fourth edition, but I may not be able to keep up with this. Webpage: http://www.math.lsa.umich.edu/~speyer/417 Linear algebra is perhaps the most important field of mathematics for computations and applications. Linear problems turn up at every step of every computation and there are well established, powerful methods to solve them. Linear methods are at the heart of computer graphics, every form of data analysis, and are the first approximation to every problem in every field of science. In this course, we will learn the computational methods, the images and the concepts of linear algebra. Grading: Your final grade will be made out of 25% Midterm 1, 25% Midterm 2, 40% Final Exam and 10% Homework. Reading: I will assign reading in the textbook. You are responsible for doing this reading in advance of class. This will allow me to use class time more efficiently to clear up points of confusion and present alternate perspectives.

Vermeer demonstrates an excellent understanding of perspective. Did he know that it could be described using matrices?

Homework Policy: You may collaberate on homework, but you must write up and turn in your own problem set, and you must disclose any people with whom you worked. Homework is due Wednesdays in class. If you cannot turn in your homework at that time, you must contact me in advance to arrange another time. You are free to seek help from me, from each other (disclosed as above), and from the tutors at the Mathlab. If you get help from someone outside this group, it should be limited in nature, and must be disclosed. You absolutely MAY NOT post homework problems to internet discussion boards.

Will Hunting realizes that he can count paths through a network using matrices. If you want to learn more about this, take Math 465 or 565.

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Date	Topics	Reading	Homework (5th edition)	Homework (4th edition)
Sept 4	Introduction to Linear Algebra
Sept 6	Solving linear equations, echelon form	Read 1.1 and 1.2
Sept 9	Examples and applications	Read 1.3
Sept 11	Matrix operations		1.1 14, 20 and 30 1.2 10, 44 and 45 (Use a calculator for 45) 1.3 4, 48 and 58	1.1 14, 26, 28 1.2 10, 42 and 43 (use a calculator for 43) 1.3 4, 48 and 58
Sept 13	Linear maps	Read 2.1, additional notes.
Sept 16	Injectivity, surjectivity and so forth	Read 2.2
Sept 18	Inverting matrices	Read 2.4 and this note	2.1 8, 12, 26, 28 2.2 20, 32 2.3 8	2.1 8, 12, 26, 28 2.2 20, 32 2.3 6
Sept 20	Linear maps in geometry	Reread 2.2
Sept 23	Subspaces	Read 3.1
Sept 25	Bases	Read 3.2	2.4 2, 6, 29, 40, 41. (In 41, as with all questions, you must justify your answers.) 3.1 6, 24, 32, 34, 37	2.4 2, 4, 29, 40, 41. (In 41, as with all questions, you must justify your answers.) 3.1 8, 24, 32, 34, 37
Sept 27	Computing bases	Read 3.3
Sept 30	Theorems about bases	Optional reading: Every subspace has a basis
Oct 2	Review		Look at review problems	Look at review problems
Oct 4	MIDTERM 1
Oct 7	Discuss Midterm, start dot products	Optional reading: Proof of the Cauchy-Schwartz inequality
Oct 9	Transpose and orthogonal complement	Read these notes on transpose	3.3 30, 38 5.1 6, 10, 16	3.3 30, 38 5.1 6, 10, 16
Oct 11	Orthogonal bases, orthogonal projection	Read 5.1
Oct 14	Fall Break
Oct 16	Computing orthogonal bases	Read 5.2	Homework delayed until Friday	Homework delayed until Friday
Oct 18	Catch up, orthogonal matrices	Read 5.3 and the formula for orthogonal projection	5.1 26, 28 5.2 4, 6, 18, 29 5.3 2, 6, 8, 10	5.1 26, 28 5.2 4, 6, 18, 29 5.3 2, 6, 8, 10
Oct 21	The method of least squares	Read 5.4
Oct 23	Intro to Determinants	Reread 2.4; read 6.1	5.4 20, 36, 38 6.1 12, 14, 24, 26, 40, 44	5.4 20, 36, 38 6.1 12, 14, 24, 26, 40, 44
Oct 25	Computing Determinants
Oct 28	Properties of Determinants	Read 6.2
Oct 30	Geometry of Determinants	Read 6.3 and Proof that determinants multiply	6.2 2, 12, 14, 38, 42 6.3 6, 18	6.2 2, 12, 14, 38, 42 6.3 6, 18
Nov 1	Review
Nov 4	MIDTERM 2
November 6	Discuss exam, introduction to Eigenvalues	Read 7.1	NO HOMEWORK	NO HOMEWORK
November 8	Introduction to eigenvalues and computing eigenvalues (Prof. Greiss substitutes)	Read 7.2
November 11	Computing eigenvalues and eigenvectors	Read 7.3
November 13	Theorems about eigenvectors	Read these notes	7.1 4, 6, 12, 16, 18 7.2 8, 12, 18 7.3 8, 10, 24	7.1 4, 6, 12, 16, 18 7.2 8, 12, 18 7.3 8, 10, 24
November 15	Complex eigenvalues	Read 7.5
November 18	Dynamical systems	Reread 7.1, read 7.4. WARNING These sections are significantly improved in the fifth edition. Please find a fifth edition copy to borrow
November 20	Eigenvalues and oscilating systems		7.1 68, 72 7.4 4, 34 7.5 20, 28	7.1 51, 54 7.3 44 7.4 32 you need not compute 7.5 20, 28
November 22	Symmetric matrices	Read 8.1. Optional reading Proof of the spectral theorem
November 25	Visualizing quadratic forms	Read 8.2
November 27	No class. You may turn in homework on Monday, or in my ofice door until Wednesday.		8.1 4, 14, 16 8.2 2, 16, 18	8.1 4, 14, 16 8.2 2, 16, 18
November 29	No class. Happy Thanksgiving!
December 2	Quadratic Forms and Optimization
December 4	Singular Values More optimization	Read 8.3	8.2 21, 22 8.3 4, 6	8.2 21, 22 8.3 4, 6
December 6	Singular Values	Read notes on minimizing quadratics subject to linear constraints
December 9	Singular value decomposition in data analysis	Just for fun, not assigned: Cosma Shalizi has excellent notes on Principal Component Analysis. Wolfram software has a great online demo on image compression.
December 11	REVIEW	I encourage you to look at the practice exams in advance of class. You also may find this topic list helpful.
December 12	Additional review, 11-12 AM, East Hall 3088
December 13	FINAL EXAM, 296 Dennison, 4-6 PM
Date	Topics	Reading	Homework (5th edition)	Homework (4th edition)