## 403 – HW 2 – Linear algebra over F_2, and orthogonality

March 3, 2014

This set is due Tuesday, March 18, at the beginning of lecture.

## Guns. II

February 26, 2014

A follow-up to http://andrescaicedo.wordpress.com/2014/02/26/guns-i/
From Bob Kustra, President of BSU:

Date: Wed, 26 Feb 2014 14:18:45 -0700
Subject: Concerns with Idaho’s Guns on Campus Bill

I know many of you have heard about the bill to remove from the State Board of Education and the administrations of public universities and colleges in Idaho the right, currently held, to prohibit weapons on their campuses. It has passed the Idaho Senate and now awaits a hearing in the House. I have very serious concerns about the bill and its implications and have spoken out against it in recent media. I should note that every public college and university president in our state and every member of the State Board of Education, with responsibility for K-20 schools in Idaho, also oppose the bill. I think it’s important for everyone in the Boise State family to know and understand my concerns about this bill.

Bob Kustra, President Boise State University

*An ‘Open Carry’ Law: This bill permits those with certain permits to carry concealed weapons on campus except in residence halls and in public entertainment facilities with seating capacity of 1000 or more. Naturally, folks are focusing on this being about concealed weapons being allowed on campus. But under Idaho law, anyone with a concealed carry permit can also openly carry a weapon.

So that means this bill would allow students and others to strap weapons openly on their hips or across their shoulders as they stroll across campus or enter their classrooms. Ironically, while they would be prohibited from carrying a concealed weapon into Bronco Stadium or the Morrison Center, we could not prohibit anyone with a permit from openly carrying their weapons into either of those venues or into Taco Bell Arena.

We have no idea how much this will affect booking entertainment and athletic events into Taco Bell Arena or conferences into the Student Union Building, but staff warns that it will surely have a chilling effect on these opportunities and revenues based on weapon-ban requirements by these groups in past booking contracts.

That is not the picture of Boise State University that any of us should need or want. The sponsor keeps saying this will not change campus life, but it surely will. What sort of message does this send about our schools and, indeed, our state? The bill also requires signage “conspicuously posted at each point of public ingress” throughout our campus implementing these changes. Again, what kind of message does this send about Boise, about Idaho and about our priorities?

Utah allows concealed weapons on campuses (most states do not) but even Utah does not permit open carrying of weapons on its public university campuses. In fact, of the handful of states that allow concealed carry on public campuses, none allow open carry of weapons as this bill allows.

*A “basic right?” Sen. Curt MacKenzie, sponsor of the bill, claims this is about restoring a basic right, which implies that anyone opposed to this bill would be opposed to and is seeking to infringe upon the rights granted in the Second Amendment. Yet the United States Supreme Court, including its most conservative members, have recognized that firearms prohibitions in “schools and government buildings” and other “sensitive areas” could well be necessary and thus never extended constitutional protection against regulations or prohibitions when schools or government buildings are involved. Justice Scalia wrote in the Heller case that “nothing in our opinion should be taken to cast doubt on the longstanding …laws forbidding the carrying of firearms in sensitive places such as schools and government buildings…” In the McDonald case, Justice Alito added that the laws prohibiting weapons in schools and government building are valid and that the Court “repeat[s] those assurances here.”

It is also interesting that when universities are given a choice, as private schools are, they most always choose to be gun free on their campuses. Two prominent examples are BYU-Idaho and Northwest Nazarene University, the latter the school from which Sen. MacKenzie is an alumnus. This bill will not affect private school policies. Those “basic rights” purported by proponents will not be available there.

*Unfunded mandate: It is currently unknown what regulations will be promulgated to guide universities in the implementation of this bill should it become law. But it is a certainty that it will lead to major expenditures in the arming and training of security forces; in the likelihood of needing metal detectors at residence hall entrances and entrances of other venues on campus where this law would prohibit concealed weapons. Early estimates from affected institutions from across the state are running into the millions of dollars with no state funding provided.

The bill likely would require such costly inspection measures because if we did not take these steps, we could be open to lawsuits for not enforcing the law and its restrictions. The immunity clause in the bill does not provide protection to the universities in those cases.

*Loss of local control: This bill strips a critically important policy decision from the members of the State Board of Education, and from the locally elected trustees of community colleges from across Idaho. It imposes central control from the State Capitol that assumes one size will fit all in this matter, when certainly we are seeing how that is not the case as each university or college is realizing its particular problems with this bill.

* Children unintended participants: Weapons, concealed or otherwise, are not allowed in Idaho’s elementary, middle and high schools. Yet children of these same ages are frequently on Boise State’s campus and cannot be kept separate from where guns would now be permitted.

The bill’s sponsor may be targeting universities in the belief that all students are aged 18 and above. This assumption misses completely our strong role and mission to serve youth of all ages. Since we have young people on our campus nearly constantly throughout the year, it is impossible to list all the occasions. For just one example, this week our Student Union is hosting the Idaho High School Student Council meetings involving 800 high school students from across the state.

It bears noting specifically that we operate a Children’s Center with 182 children annually ages 2 months to 6 years on campus.

In the summer, our campus is alive with young people participating in athletic and academic camps. During this time, they are all over campus including the Student Union and, often, residence halls. A partial list follows: Summer Chamber Music Camp; e-Camp; football, swim, volleyball, soccer, tennis, softball, cross country, lacrosse, gymnastics and wresting camps; DanceFest; Adventure Program, Youth Sports Program, Elementary-Level Academy, Literacy Academy, Morrison Center Performance Camps, Teen GameLab Design Camp, STEM Summer Adventure.

*Law enforcement concerns: There are good reasons that the police force that provides our campus security is opposed to this bill. Boise Police Chief Mike Masterson was prohibited from offering testimony in the Senate State Affairs Committee chaired by the bill’s sponsor, but if he had been permitted to testify, he had planned to focus on the vast gulf in training between constantly drilled police officers and the “enhanced” concealed weapons permit holders, who go through one 8-hours class once every five years. He and other law enforcement leaders have pointed out the difficulty in having armed “good guys” and armed “bad guys” as law enforcement comes upon an emergency scene. It will be almost impossible for them to sort it out correctly and tragedy could well be the consequence if they cannot.

His prepared testimony included the following: “I’m here to oppose this legislation and am joined by virtually all police chiefs across the state policing Idaho’s college campuses as well as presenting a letter from Chief Dan Hall, president of the Idaho Chiefs of Police Association, opposing it as well.”

## Guns. I

February 26, 2014

From Bob Kustra, president of BSU:

Date: Tue, 4 Feb 2014 16:27:56 -0700
Subject: Guns on Campus

Dear Boise State students, faculty, staff, parents and friends,

The Idaho Legislature is considering a new bill that would allow guns on campus and take some authority to regulate weapons away from the State Board of Education and Idaho’s public universities and colleges. As President of Boise State, I have joined with all of Idaho’s public university and college presidents in opposition to this bill.

I want to share with you how we and the leaders of Idaho’s public universities and colleges work closely with our security personnel and local law enforcement agencies to assure our students a safe learning environment. After the shootings at Virginia Tech and Northern Illinois University, we reported to our trustees on the State Board on our security plans to keep our campuses safe and we believe that we have taken the necessary precautions to guarantee our students’ safety to the very best of our ability. Embedded in an 80-page Emergency Operations Plan for Boise State are two key elements — the emergency alert sent out immediately when an active shooter is confirmed and 30-second to 2-minute response time by the Boise Police who patrol the campus. We take this responsibility as our most important duty to our students, our faculty and our guests on our campuses. And we feel strongly that our campus security officers and our local law enforcement officers are in the best position to manage campus security, not a state law that does not benefit from the actual law enforcement experience of our local police force.

In fact, we can find no recorded incident in which a victim—or a spectator—of a violent crime on a campus has prevented a crime by brandishing a weapon. In fact, professional law enforcement officials claim that increasing the number of guns on a campus would increase police problems and make it difficult for police officers in a shooting situation to tell the good shooter from the bad shooter and inadvertently shoot an innocent person. Weapons on campus may, in fact, lead to an acceleration of conflict in stressful situations.

If you have questions or concerns about the proposed bill, Senate Bill 1254
http://legislature.idaho.gov/legislation/2014/S1254.pdf,
I encourage you to contact the members of the Senate State Affairs Committee, which will be considering the bill soon. The Senate State Affairs Committee can be reached at
sstaf@senate.idaho.gov.
These are the members of the committee:

Chairman Curt McKenzie, R, District 13, Nampa
Sen.  Bart Davis, R, District 33, Idaho Falls
Sen.  Russell Fulcher, R, District 22, Meridian
Sen. Brent Hill, R, District 34, Rexburg
Sen. Chuck Winder, R, District 20, Boise
Sen.  Patti Anne Lodge, R, District 11, Huston
Sen.  Jeff Siddoway, R, District 35, Terreton
Sen. Michelle Stennett, D, District 26, Ketchum
Sen. Elliot Werk, D, District 17, Boise

If this bill is approved by the Senate Committee, it will then travel to the full Senate for a vote. We urge you to consider how you feel about this measure and to contact your own Senator to record your position on the bill.  Messages can be left by calling
(208) 332-1000
or by accessing email addresses at:
http://legislature.idaho.gov/senate/membership.cfm

Sincerely,

Bob Kustra
President

## Posets and Lattices

February 4, 2014

Definition 1.
A lattice is a nonempty poset with a maximum and a minimum, and such that any nonempty finite subset has a supremum (or join) and an infimum (or meet).

One could just as well simply say that a lattice is a poset $\mathcal L$ where any finite subset has a supremum and an infimum, using the (natural) convention that the supremum of the empty set is the minimum of $\mathcal L$, and its infimum is the maximum of $\mathcal L$.

The requirement that a lattice $\mathcal L$ has a minimum and a maximum is not universally followed, and lattices with this additional property are sometimes called bounded.

Examples 2.
Any linearly ordered set with a minimum and a maximum is a lattice. In this case, all infima and suprema of nonempty finite sets are just minima and maxima.

For a more interesting example, consider $\mathbb N$, partially ordered by divisibility. This is a lattice, with minimum $1$ and maximum $0$. The infimum of a nonempty set $X\ne\{0\}$ is the greatest common divisor of the elements of $X$, and its supremum is their least common multiple.

Given a set $X$, its power set $\mathcal P(X)$ is a lattice under containment, $\subseteq.$ In fact, in this case any nonempty subcollection of $\mathcal P(X)$ (whether finite or not) has an infinimum (its intersection) and a supremum (its union). This is an example of a complete lattice, as in Definition 3.

Given a set $X$, let $\mathrm{Fin}(X)$ denote the collection of finite subsets of $X$, partially ordered under containment. If $X$ is infinite, $\mathrm{Fin}(X)$ is not a lattice (it lacks a maximum), but $\{X\}\cup\mathrm{Fin}(X)$ is.

The partition lattice $\Pi_n$ has as elements the partitions of $\{1,\dots,n\}$ where, as usual, a partition of a set $A$ is a collection $\mathcal A$ of pairwise disjoint nonempty subsets of $A$ whose union is $A$. The set $\Pi_n$ is ordered by refinement: If $x,y$ are partitions in $\Pi_n$, we say that $x\le y$ iff every set in $x$ is a subset of a set in $y$.

Definition 3.
A lattice $\mathbb P=(P,\le)$ is complete iff any nonempty subset of $P$ has a supremum and an infimum.

With the same convention as above, we could as well say that any subset of $P$ admits a supremum and an infimum.

Examples 4.
By the completeness of the real line, any compact interval ${}[a,b]\subseteq\mathbb R$ is a complete lattice under the usual order.

$(\mathbb N,|)$ is actually a complete lattice. The supremum of any infinite set is $0$.

The collection of convex subsets of $\mathbb R^2$ is a complete lattice under containment. The infimum  of a nonempty family $\mathcal F$ of convex sets is just their intersection. Its supremum is their convex hull, that is, the intersection of all the convex subsets of $\mathbb R^2$ that contain all sets in $\mathcal F$.

Given a group $G$, the collection $\mathrm{Sub}(G)$ of subgroups of $G$, partially ordered by containment, is a complete lattice.

If $X$ is a linearly ordered set with a supremum and an infimum, then $X$ is a complete lattice iff it is Dedekind complete, meaning that any cut admits a supremum. Cuts are defined as a natural generalization of the usual concept for $\mathbb Q$: A cut of $X$ is a pair $(A,B)$ of subsets of $X$ such that $B$ is the collection of upper bounds of $A$, and $A$ is the collection of lower bounds of $B$. Note that this notion makes sense even if $X$ is just a partially ordered set. Given a cut $(A,B)$ in $X$, we say that it admits a supremum iff $\sup(A)$ exists.

Given any partially ordered set $P$, there is always a complete lattice that contains $P$, for example, the collection of downward closed subsets of $P$, under containment, where $p\in P$ is identified with $\{x\in P\mid x\le p\}$.

Exercise 5 (MacNeille).
In fact, any poset $\mathbb P=(P,\le)$ admits a minimal completion $\mathcal L$, in the sense that $\mathcal L$ is a complete lattice containing $\mathbb P$, and $\mathcal L$ embeds (as a poset) into any complete lattice that contains $\mathbb P$ (with the embedding fixing $P$ pointwise).

As with the rationals and the reals, a natural way of defining this minimal completion $\mathcal L$ of $\mathbb P$ is simply to take as $\mathcal L$ the collection of all cuts in $\mathbb P$, partially ordered by saying that $(A,B)\le(C,D)$ iff $A\subseteq C$. This construction is called the Dedekind-MacNeille (or normal) completion, first introduced in MacNeille [Mac37]. Verify that it indeed results in the minimal completion of $\mathbb P$.

Suppose that $(A,B)$ is a cut of $\mathbb P$. Show that either $A$ and $B$ are disjoint, or else they meet at exactly one point.

Suppose that $a\in P$, $A\subseteq\mathbb P$, and $a=\max(A)=\sup(A\setminus\{a\})$. Let $\hat A$ be the downward closure of $A$, and let $B$ be the set of upper bounds of $A$. Show that $(\hat A,B)$ is a cut but $(\hat A\setminus\{a\},B)$ is not. This is slightly different from the situation with $\mathbb Q$ and its completion $\mathbb R$, where commonly we require the two sets in a cut to have no elements in common. Does the Dedekind-MacNeille completion of $\mathbb Q$ actually coincide with $\mathbb R$?

Exercise 6 (Birkhoff).
Say that a lattice $\mathcal L$ is distributive iff $p \land (q \lor r) = (p \land q) \lor (p \land r)$ for any $p,q,r\in\mathcal L$, where $a \land b$ denotes the meet of $a$ and $b$, and $a\lor b$ denotes their join.

For example, $\mathcal P(X)$ is a distributive lattice. Give an example of a lattice that is not distributive.

Prove Birkhoff’s representation theorem, also called the fundamental theorem for finite distributive lattices, stating that any such lattice is isomorphic to a lattice of finite sets, where the lattice operations are just given by union and intersection.

References

[Mac37]
Holbrook M. MacNeille.
Partially ordered sets.
Trans. Amer. Math. Soc. 42 (3), (1937), 416–460.
MR1501929.

## 403 – HW 1 – Recurrence relations

February 3, 2014

This set is due Tuesday, February 18, at the beginning of lecture.

## 314 – Foundations of Analysis – Syllabus

January 20, 2014

Math 314: Foundations of Analysis.

Andrés E. Caicedo.
Contact Information: See here.
Time: TTh 12:00 – 1:15 pm.
Place: Mathematics Building, Room 139.
Office Hours: Th, 1:30 – 3:00 pm, or by appointment. (If you need an appointment, email me a few times/dates that may work for you, and I’ll get back to you).

Textbook: Stephen Abbott. Understanding Analysis. Springer-Verlag, Undergraduate Texts in Mathematics, 2001; 257 pp. ISBN-10: 0387950605. ISBN-13: 978-0387950600.

Here is the publisher’s page. Additional information is available from the author’s page. Review (MR1807438 (2001m:26001)) by Robert Gardner Bartle at MathSciNet. Review by Jeffrey Nunemacher at the American Mathematical Monthly, Vol. 118, No. 2 (February 2011), pp. 186-189.

I will mention additional references, and provide handouts of additional material, as needed.

Contents: The department’s course description reads:

The real number system, completeness and compactness, sequences, continuity, foundations of the calculus.

I strongly suggest you read the material ahead of our meetings, and work on it frequently. You may find some of the topics challenging. If so, here is some excellent advice by Faulkner (from an interview at The Paris Review):

Personally, I find the topics we will study beautiful, and I hope you enjoy learning it as much as I did.

Please bookmark this post. I update it frequently with detailed week-to-week descriptions.

Detailed day to day description and homework assignments. All problems are from Abbott’s book unless otherwise explicitly specified:

• January 21 – 30. Chapter 1. The real numbers. Irrationality. Completeness. Countable and uncountable sets.
• January 21. Functions. Mathematical induction and the well-ordering principle.
• January 23. Sets, logic, quantifiers. Completeness.
• January 28. Completeness. Countable and uncountable sets. I recommend you read Errol Morris‘s essay on Hypassus of Metapontum, the apparent discoverer of the irrationality of $\sqrt2$.
• January 30. Comparing infinities. Counting the rationals. I recommend the following two papers on this topic: 1 and 2. Office hours this week will be on Friday, 11:45-1:15.

Homework set 1 (Due February 4). Exercises 1.2.1, 1.2.2, 1.2.7, 1.2.8, 1.2.10; 1.3.21.3.9; 1.4.21.4.7, 1.4.11 1.4.13; 1.5.3, 1.5.4, 1.5.9. See below for the required format.

• February 4 – 20. Chapter 2. Sequences and series. Limits. Cauchy sequences. Infinite series. Riemann’s rearrangement theorem.
• February 4. Rearrangements of infinite series, limits of sequences. Homework 1 is due today.
• February 6. Limit theorems.
• February 11. Limit theorems continued. Infinite series.
• February 13. Monotone convergence. The Bolzano-Weierstrass theorem.
• February 18. The Bolzano-Weierstrass theorem continued. Absolute and conditional convergence. Cauchy sequences.
• February 20. Riemann’s rearrangement’s theorem, and extensions (see here and here). The interesting paper by Marion Scheepers mentioned on the second of those links can be found here.
• Additional topics: Products of series. Double series. Formal power series and applications in combinatorics. I recommend the nice paper by Ivan Niven on this topic.

Homework set 2 (Due February 25). Exercises 2.2.1, 2.2.2, 2.2.5, 2.2.7, 2.3.2, 2.3.3, 2.3.6, 2.3.7, 2.3.9, 2.3.11, 2.4.2, 2.4.4, 2.4.5, 2.5.3, 2.5.4, 2.6.1, 2.6.3, 2.6.5, 2.7.1, 2.7.4, 2.7.6, 2.7.9, 2.7.11. See below for the required format.

• February 25 – March 6. Chapter 3. Basic topological notions: Open sets. Closed, compact, and perfect sets. The Cantor set. Connectedness. The Baire category theorem.
• February 25. The Cantor set. Open and closed sets.
• February 27. Open and closed sets, continued. Extra credit problem: Find a set of reals such that we can obtain $14$ different sets by applying to it (any combination of) the operations of complementation and closure. Kuratowski showed that $14$ is the largest number that can be obtained that way, you are welcome to also try to show that.
• March 4. Open covers, compact sets. Perfect sets. Connectedness.
• March 6. The Baire category theorem.
• Additional topics: The study of closed sets of reals naturally leads to the Cantor-Bendixson derivative, and the Cantor-Baire stationary principle. A nice reference is Alekos Kechris‘s book, Classical descriptive set theory. For the Baire category theorem and basic applications, I recommend the beginning of John Oxtoby‘s short book, Measure and category. See also the nice paper Subsum Sets: Intervals, Cantor Sets, and Cantorvals by Zbigniew Nitecki, downloadable at the arXiv.

Homework set 3 (Due March 11). Exercises 3.2.1, 3.2.3, 3.2.7, 3.2.9, 3.2.11, 3.2.12, 3.2.14, 3.3.2, 3.3.43.3.7, 3.3.9, 3.3.10, 3.4.2, 3.4.4, 3.4.5, 3.4.73.4.10, 3.5.43.5.6.

Grading: Based on homework. There will also be a group project, that will count as much as two homework sets. I expect there will be no exams, but if we see the need, you will be informed reasonably in advance.

There is bi-weekly homework, due Tuesdays at the beginning of lecture; you are welcome to turn in your homework early, but I will not accept homework past Tuesdays at 12:05 pm, or grant extensions. The homework covers some routine and some more challenging exercises related to the topics covered in the past two weeks (roughly, one homework set per chapter). It is a good idea to work daily on the homework problems corresponding to the material covered that day.

You are encouraged to work in groups and to ask for help. However, the work you turn in should be written on your own. Give credit as appropriate: Make sure to list all books, websites, and people you collaborated with or consulted while working on the homework. If relevant, indicate what software packages you used, and include any programs you may have written, or additional data.

Your homework must follow the format developed by the mathematics department at Harvey Mudd College. You will find that format at this link. If you do not use this style, unfortunately your homework will be graded as 0. In particular, please make sure that what you turn in is not your scratch work but the final product. Include partial attempts whenever you do not have a full solution.

I may ask you to meet with me to discuss details of sets, and I suggest that before you turn in your work, you make a copy of it, so you can consult it if needed.

I post links to supplementary material on Google+. Circle me and let me know if you are interested, and I’ll add you to my Analysis circle.

## 403/503 – Advanced linear algebra – Syllabus

January 19, 2014

Math 403/503: Advanced linear algebra.

Andrés E. Caicedo.
Contact Information: See here.
Time: TTh 10:30 – 11:45 am.
Place: Mathematics Building, Room 124.
Office Hours: Th, 1:30 – 3:00 pm, or by appointment. (If you need an appointment, email me a few times/dates that may work for you, and I’ll get back to you).

Textbook: Although it is not a textbook per se, our main reference will be

• Jiří Matoušek. Thirty-three Miniatures: Mathematical and Algorithmic Applications of Linear Algebra. American Mathematical Society, Student Mathematical Library, vol. 53, 2010; 182 pp. ISBN-10: 0-8218-4977-8. ISBN-13: 978-0-8218-4977-4.

Here is the publisher’s page. A preliminary version is available from the author’s page. Review (MR2656313 (2011f:15002)) by Torsten Sander at MathSciNet.

Another useful reference in the same spirit is

• László Babai, and Péter Frankl. Linear Algebra Methods in Combinatorics. With Applications to Geometry and Computer Science. Preliminary Version 2 (September 1992), 216 pages.

This book is unpublished. A copy can be obtained from the Department of Computer Science at the University of Chicago, or elsewhere. There are also several sets of lecture notes on this topic available online. See for example here or here.

We will not restrict our lectures to topics related to these applications, and also cover some more traditional material, and some numerical aspects of the theory. A pdf of Uwe Kaiser‘s notes can be found here and the TeX source here. I will provide handouts of additional material as needed.

Contents: The department’s course description reads:

Concepts of linear algebra from a theoretical perspective. Topics include vector spaces and linear maps, dual vector spaces and quotient spaces, eigenvalues and eigenvectors, diagonalization, inner product spaces, adjoint transformations, orthogonal and unitary transformations, Jordan normal form.

The way we will develop the theory is by studying examples of some of its typical applications, and then covering the topics needed to understand these examples.

Please bookmark this post. I update it frequently with detailed week-to-week descriptions.

Detailed day to day description and homework assignments:

• January 21. Review of basic linear algebra: Vector spaces, fields. On “field” vs “body”, see here.
• January 23. Linear transformations, matrices, bases. Quick overview of topics. Solving linear recurrences (Matoušek’s lectures 1 and 2).
• January 28. Recurrent sequences. A useful reference (unfortunately in Spanish): Sucesiones recurrentes, by A. I. Markushevich. The goal is to use these results to motivate the study of the Jordan canonical form. Office hours this week will be on Friday, 11:45-1:15.
• January 30. Recurrent sequences. Diagonalizable and non-diagonalizable matrices.
• February 4. Ideals. Minimal polynomials. Homework 1, due February 18.
• February 6. Jordan blocks. Recurrent sequences.
• February 11. The Jordan form theorem.
• February 13. Odds and ends. Other possible approaches to solving linear recurrences. (We will revisit generalized eigenspaces, direct sums, and the Jordan form theorem later on.) Next topic: Parity (including Matoušek’s lectures 3, 4, 17).
• February 18. Parity. Oddtown. (Matoušek’s lecture 3).
• February 20. The Eventown theorem. Bilinear forms, inner products, isotropic vectors, singular spaces. (Section 2.3 in the Babai-Frankl notes.)
• February 25. Singular spaces (continued).
• February 27. The Berlekamp-Graver strong Eventown theorem.
• March 4. The generalized Fisher inequality and additional results on set systems with forbidden intersections (Matoušek’s lectures 4 and 17). Finite projective planes. Homework 2, due March 25. Next topic: Finding eigenvalues.

Lecture notes (version of 02/12/14).

Grading: Based on homework. No extensions will be granted, and no late work will be accepted. I expect there will be no exams, but if we see the need, you will be informed reasonably in advance. You are encouraged to work in groups and to ask for help. However, the work you turn in should be written on your own. Give credit as appropriate: Make sure to list all books, websites, and people you collaborated with or consulted while working on the homework. If relevant, indicate what software packages you used, and include any programs you may have written, or additional data.

I may ask you to meet with me to discuss details of sets, and I suggest that before you turn in your work, you make a copy of it, so you can consult it if needed.

I post links to supplementary material on Google+. Circle me and let me know if you are interested, and I’ll add you to my Linear Algebra circle.