Math 101: Graduate Linear Algebra
Fall 2017
Course Info:
- Lectures: Monday, Wednesday, Friday, block 11 (11:30 a.m. - 12:35 p.m.)
- x-period: Tuesday, block 11X (12:15-1:05 p.m.)
- Dates: 11 September 2017 - 14 November 2017
- Room: Kemeny 201
- Instructor: John Voight
- Office: 341 Kemeny Hall
- E-mail: jvoight@gmail.com
- Office hours: Tuesdays, 1:15-2:15 p.m. or by appointment
- Course Web Page: http://www.math.dartmouth.edu/~m101f17/
- Prerequisites: A previous course in undergraduate algebra is strongly recommended.
- Required Text: David S. Dummit and Richard M. Foote, Abstract Algebra, 3rd ed., Wiley, 2003; see their errata.
- Recommended Texts:
- Stephen Friedberg, Arnold Insel, and Lawrence Spence, Linear algebra, 4th. ed., Pearson, 2002.
- Steven Roman, Advanced linear algebra, 3rd. ed., GTM vol. 135, 2007.
- Serge Lang, Algebra, 3rd. ed., GTM vol. 211, Springer-Verlag, 2005.
- Thomas Hungerford, Algebra, 8th ed., GTM vol. 73, Springer-Verlag, 2003.
- Paolo Aluffi, Algebra: Chapter 0, GSM, American Mathematical Society, 2009.
- Michael Artin, Algebra, 2nd. ed., Pearson, 2010.
- Grading: Grade will be based on daily homework (25%), weekly homework (25%), a midterm (25%), and a final exam (25%).
Syllabus:
We follow the official Math 101 syllabus, but we will not cover group theory.
[PDF] Syllabus
| Linear algebra | 1 | 11 Sep | (M) | Introduction; Definitions and basic theory | DF section 11.1 FIS chapter 1 | WS 1 [TeX] [PDF] Daily HW 1 [TeX] [PDF] |
| 2 | 12 Sep | (T) | Infinite-dimensional vector spaces | DF appendix I.2 FIS section 1.7 |
|
| 3 | 13 Sep | (W) | The matrix of a linear transformation | DF section 11.2 | Daily HW 3: DF 11.2.11 |
| 4 | 15 Sep | (F) | Dual vector spaces | DF section 11.3 FIS section 2.6 | Daily HW 4 [TeX] [PDF] Weekly HW 1 [TeX] [PDF] Weekly HW 1 solutions [TeX] [PDF] |
| 5 | 18 Sep | (M) | Annihilators | FIS section 2.6 Roman pp. 101-107 | Daily HW 5 [TeX] [PDF] |
| 6 | 20 Sep | (W) | Tensor products | Roman chapter 14 | Daily HW 6 [TeX] [PDF] |
| 7 | 22 Sep | (F) | Tensor products and bilinear forms | Roman chapter 14 | Weekly HW 2 [TeX] [PDF] Weekly HW 2 solutions [TeX] [PDF] |
| - | 25 Sep | (M) | No class: JV in Banff | ||
| - | 27 Sep | (W) | No class: JV in Banff | ||
| 8 | 29 Sep | (F) | Adjoints, self-adjoint operators, inner products | FIS sections 6.1-6.3 | Daily HW 8 [TeX] [PDF] |
| 9 | 2 Oct | (M) | Orthogonal, normal, and unitary operators | FIS sections 6.4-6.5 | Daily HW 9 [TeX] [PDF] |
| 10 | 3 Oct | (T) | Singular value decomposition (SVD) | FIS section 6.7 | |
| 11 | 4 Oct | (W) | Orthogonal projections and the spectral theorem | FIS section 6.6 | Daily HW 11 [TeX] [PDF] Weekly HW 3 [TeX] [PDF] Weekly HW 3 solutions [TeX] [PDF] |
| Modules | |||||
| 12 | 6 Oct | (F) | Basic definitions and examples | DF section 10.1 | Daily HW 12 [TeX] [PDF] |
| 13 | 9 Oct | (M) | Quotient modules and module homomorphisms | DF section 10.2 | Daily HW 13 [TeX] [PDF] |
| 14 | 11 Oct | (W) | Generation of modules, direct sums, and free modules | DF section 10.3 | |
| - | 12 Oct | (R) | Midterm exam, covering linear algebra (4:30-6:30 p.m., Kemeny 343) | Midterm exam solutions [TeX] [PDF] | |
| 15 | 13 Oct | (F) | Tensor products of modules | DF section 10.4 | |
| 16 | 16 Oct | (M) | Tensor products, exact sequences | DF section 10.5 | Daily HW 16 [TeX] [PDF] Weekly HW 4 [TeX] [PDF] Weekly HW 4 solutions [TeX] [PDF] |
| 17 | 18 Oct | (W) | Diagram chases | DF section 10.5 | Daily HW 17 [TeX] [PDF] |
| 18 | 20 Oct | (F) | Projective modules | DF section 10.5 | Daily HW 18 [TeX] [PDF] |
| 19 | 23 Oct | (M) | Injective modules; rings of fractions | DF section 10.5 | Daily HW 19 [TeX] [PDF] |
| 20 | 25 Oct | (W) | Localization | DF sections 7.5, 15.4 | Daily HW 20 [TeX] [PDF] Weekly HW 5 [TeX] [PDF] Weekly HW 5 solutions [TeX] [PDF] |
| 21 | 27 Oct | (F) | Localization and locally free modules | Daily HW 21 [TeX] [PDF] | |
| Modules over PIDs, canonical forms | |||||
| 22 | 30 Oct | (M) | Euclidean domains and PIDs | DF sections 8.1-8.2 | Daily HW 22 [TeX] [PDF] |
| 23 | 31 Oct | (T) | UFDs, FTFGMPID | DF sections 8.1-8.2 DF section 12.1 |
|
| 24 | 1 Nov | (W) | Basic theory (FTFGMPID) | DF section 12.1 | |
| 25 | 3 Nov | (F) | Smith normal form | Daily HW 25 [TeX] [PDF] Weekly HW 6 [TeX] [PDF] Weekly HW 6 solutions [TeX] [PDF] | |
| 26 | 6 Nov | (M) | Rational canonical form | DF section 12.2 | Daily HW 26 [TeX] [PDF] |
| 27 | 7 Nov | (T) | Noetherian rings | ||
| 28 | 8 Nov | (W) | Jordan canonical form, applications | DF section 12.3 | |
| Category theory and wrap-up | |||||
| 29 | 10 Nov | (F) | Categories and functors | DF Appendix II | Daily HW 29 [TeX] [PDF] |
| 30 | 13 Nov | (M) | Natural transformations, wrap-up | ||
| - | 17 Nov | (F) | Final exam, comprehensive (8:00 a.m.-11:00 a.m.) in Kemeny 108 | Final exam solutions [TeX] [PDF] | |
Homework:
The homework assignments will be assigned on a daily basis and weekly basis and will be posted above.
Daily homework is due the following class period: we will discuss the problem in class, and you will self-assess in red pen. At the end of the term, all daily homework will be collected, with a short concluding self-assessment.
Weekly homework is due as indicated, and will collected and graded in the usual manner.
Cooperation on homework is permitted (and encouraged), but if you work together, do not take any paper away with you--in other words, you can share your thoughts (say on a blackboard), but you have to walk away with only your understanding. In particular, you must write the solution up on your own. Please acknowledge any cooperative work at the end of each assignment.
Plagiarism, collusion, or other violations of the Academic Honor Principle, after consultation, will be referred to the The Committee on Standards.
[PDF] Homework Submission Guidelines