Linear Algebra

Jacobs University, Fall 2018

News

The tutorial on Tuesday, Dec. 4, starts at 15:45.
Final on Wed, Dec. 12, see below for more details.
The tutorial on Tuesday, Nov. 13, has to be canceled. Homework sheets 7 and 8 are discussed together in the tutorial on Nov. 20.
Midterm on Wed, Oct. 24, see below for more details.
The tutorial on Tuesday, Oct. 2, has to be canceled. Homework sheets 3 and 4 are discussed together in the tutorial on Oct. 9.
The first tutorial will be on Tuesday, Sep. 18, from 17:15 - 18:30 in room 120 in Research I.

Contact Information

Instructor: Prof. Sören Petrat

Email: s.petrat AT jacobs-university.de

Office: 112, Research I

Office hours: drop by my office anytime, or write me an email before to make sure I'm there

Teaching Assistant: Khadeeja Afzal

Email: k.afzal AT jacobs-university.de

Office hours: by email appointment

Time and Place

Class: Mon 9:45 - 11:00, West Hall 4; Wed 8:15 - 9:30, West Hall 4

Tutorial Session: Tue 15:45 - 17:00, Room 120, Research I

Syllabus

Syllabus (as of Sep. 1, 2018) available here. (Note that the Syllabus will not be updated, the most recent information can be found on this website.)

Textbooks

The class material is similar to the following textbooks:

A. I. Kostrikin and Y. I. Manin - Linear Algebra and Geometry (Gordon and Breach Science Publishers). This class covers parts of Chapters 1 and 2.
S. Axler - Linear Algebra Done Right (Springer, third edition).

Grading

See the Syllabus.

Exams

There will be two exams, a midterm and a final. The midterm will cover the material from the first half of the course and the final will cover all material with emphasis on the second half. Note that this class uses gradescope for grading exams, see here for more information.

This year's exams can be found here: Midterm, Final.

Midterm Exam: Wed, Oct. 24, 2018, 8:15 - 9:30, West Hall 4

The exam begins at 8:15 sharp, so please come a few minutes earlier to be seated at 8:15. The midterm is about all material up to the Oct. 10 session. You only need to bring a pen, no calculators or notes! Please be aware of the University policy regarding missing exams: Policies for Undergraduate Studies. In particular, in case you are missing an exam due to illness (only with doctor's note) or an emergency, you have to notify me (by email) before the beginning of the exam!

Final Exam: Wed, Dec. 12, 2018, 12:30 - 14:30, Lecture Hall Research II

The exam begins at 12:30 sharp, so please come a few minutes earlier to be seated at 12:30. The final is about all material up to and including the Dec. 5 session, with an emphasis on the material of part II of this course. You only need to bring a pen, no calculators or notes! Please be aware of the University policy regarding missing exams: Policies for Undergraduate Studies. In particular, in case you are missing an exam due to illness (only with doctor's note) or an emergency, you have to notify me (by email) before the beginning of the exam!

Homework Sheets

Each week on Monday (with exceptions) there will be a homework assignment. These are an integral part of the coursework and working on the exercise sheets consistently is the best preparation for the exams. Note that

The two worst homework sheets are not considered for grading.
It is encouraged to discuss the homework sheets with your classmates (e.g., discuss how to come up with the solution or what the right way of approaching the problem is). On the other hand, the solutions must be written down and handed in individually! Copying the solutions from somebody else is a violation of Academic Integrity.

Homework Sheets are to be handed in individually at the beginning of class. Alternatively, they can be deposited in a mailbox at the main entrance of Research I at some point before class on Monday (right next to the entrance there is a mailbox labeled "Linear Algebra"). No late hand ins except with a doctor's note that proves you were sick several days during the time you had working on the homework sheet!

Date	Sheet Number	Due Date
Sep. 12, 2018	Sheet 1	Sep. 17, 2018
Sep. 17, 2018	Sheet 2	Sep. 24, 2018
Sep. 24, 2018	Sheet 3	Oct. 01, 2018
Oct. 01, 2018	Sheet 4	Oct. 08, 2018
Oct. 08, 2018	Sheet 5	Oct. 15, 2018
Oct. 29, 2018	Sheet 6	Nov. 05, 2018
Nov. 05, 2018	Sheet 7	Nov. 12, 2018
Nov. 12, 2018	Sheet 8	Nov. 19, 2018
Nov. 19, 2018	Sheet 9	Nov. 26, 2018
Nov. 26, 2018	Sheet 10	Dec. 03, 2018

Class Schedule

Will be updated while class is progressing.

Below, please click on the date to download the lecture notes of this day.

(Note that the book references given below offer only a rough orientation. Sometimes, only parts of a particular chapter are covered in class.)

Date	Topics
Sep. 05, 2018	Vector spaces Kostrikin/Manin Ch. 1 §1; Axler (third edition) Ch. 1.A, 1.B
Sep. 10, 2018	Subspace, span, basis parts of Kostrikin/Manin Ch. 1 §2; Axler Ch. 1.C, 2.A, 2.B
Sep. 12, 2018	Basis and dimension parts of Kostrikin/Manin Ch. 1 §2; Axler Ch. 2.B, 2.C
Sep. 17, 2018	Zorn's lemma, Linear Operators, Dual Space Kostrikin/Manin Ch. 1 §1.9, §2.17-2.20 and §3.1-3.5; Axler Ch. 3.A and beginning of 3.F
Sep. 19, 2018	Isomorphisms Kostrikin/Manin Ch. 1 §3.6-3.10; Axler beginning of Ch. 3.D
Sep. 24, 2018	Dual map; Kernel, image and dimension; Matrices Kostrikin/Manin Ch. 1 §7.4, §3.11-3.13, §4.4-4.7; Axler parts of Ch. 3.F (3.99-3.101), Ch. 3.B
Sep. 26, 2018	Matrices; Sums, direct sums and dimensions Kostrikin/Manin Ch. 1 §4.4-4.9, §5.2-5.3; Axler Ch. 3.C, the matrix parts of Ch. 3.D, Ch. 1.C, and 2.43 in Ch. 2.C; if you like you can also look around in Axler Ch. 10 a bit to get more background on the trace and determinant (most of which should be familiar to you)
Oct. 01, 2018	Sums, direct sums, arrangements, and projections Kostrikin/Manin Ch. 1 §5
Oct. 03, 2018	German Unity Day, no classes
Oct. 08, 2018	Quotient spaces; Fundamental spaces of a linear operator Kostrikin/Manin Ch. 1 §6; Axler parts of Ch. 3.E
Oct. 10, 2018	Fundamental spaces of a linear operator Kostrikin/Manin Ch. 1 §6.10, §7.6; Axler parts of Ch. 3.F
Oct. 15, 2018	Linear operators on finite dimensional spaces (diagonalizability, characteristic polynomial) Kostrikin/Manin Ch. 1 §8 (most parts); Axler Ch. 5 (covers much of the same topics in a slightly different way)
Oct. 17, 2018	Diagonalizability, Cayley-Hamilton, generalized eigenspaces Kostrikin/Manin Ch. 1 §8 (most parts); Axler Ch. 8.C (Axler proves the abstract Jordan decomposition first and uses it to prove Cayley-Hamilton)
Oct. 22, 2018	Reading Day, no classes
Oct. 24, 2018	Midterm Exam
Oct. 29, 2018	Jordan decomposition and Jordan basis Kostrikin/Manin Ch. 1 §8.9, §9.1-9.6; Axler most parts of Ch.s 8.A, 8.B (only the first part)
Oct. 31, 2018	Day of Reformation, no classes
Nov. 05, 2018	Jordan normal form Kostrikin/Manin Ch. 1 §9; Axler Ch. 8.D
Nov. 07, 2018	(De)complexification most parts of Kostrikin/Manin Ch. 1 §12; Axler Ch. 9.A is about the same topic but different aspects are covered
Nov. 12, 2018	Bilinear forms, inner products, and their symmetry Kostrikin/Manin Ch. 2 §2.1-2.5; some of the material can also be found in Axler Ch.s 6 and 7
Nov. 14, 2018	Classification of forms for dimension 1 or 2 Kostrikin/Manin Ch. 2 §2.5-2.10 and 3.1
Nov. 19, 2018	General classification of bilinear forms Kostrikin/Manin Ch. 2 §3.1-3.4
Nov. 21, 2018	General classification of bilinear forms Kostrikin/Manin Ch. 2 §3.5-3.8
Nov. 26, 2018	Quadratic Forms, orthogonalization algorithms and orthogonal polynomials Kostrikin/Manin Ch. 2 §3.9 and 4
Nov. 28, 2018	Euclidean and unitary spaces Kostrikin/Manin Ch. 2, most parts of §5 and 6
Dec. 03, 2018	Orthogonal and Unitary operators Kostrikin/Manin Ch. 2, most parts of §7
Dec. 05, 2018	Self-adjoint operators Kostrikin/Manin Ch. 2, most parts of §8
Dec. 12, 2018	Final Exam