## Math 202A (Fall 2021)

### Summary

This is the first installment of the graduate Applied Algebra sequence, and concerns linear algebra with a particular focus on matrix analysis / linear analysis.

In other words, we are interested in the intersection between familiar linear algebra concepts (essentially, solving systems of linear equations), and quantitative or analytic notions of size, errors and approximations. These are essential questions when applying linear algebra over the real or complex numbers to concrete problems.

We do not focus heavily on numerical algorithms or techniques but this is essential background for those subjects. The course is strongly proof-based, and I will try to emphasize the relationship between these proofs and practical implementations.

### Logistics

**Lectures** are held on Mondays, Wednesdays and Fridays, 1400—1450 in AP&M–2402.
Note there is *no* lecture on Wednesday November 24 or Friday November 26 to allow for Thanksgiving.

Lectures will be podcast, but students are generally expected to attend lecture unless there is a particular reason not to. If you are likely to be absent from a significant number of lectures, please contact the instructor.

The lecturer's office hours will be split between in-person and virtual. The TA's office hours are will be hybrid in-personal / virtual. All this is subject to change. See the calendar below for times.

There will be a **midterm exam** on Wednesday 27th October, during usual class time.
There is a **final exam** on Wednesday December 8, 1500—1759.

**Homework** will be set every week, due by 2359 each Sunday, starting at the end of week 1 (so, 2359 on Sunday October 3). Work should be handed in via Gradescope.

The **grade breakdown** is 25% for the midterm, 30% for homework and 44.5% for the final exam.
The remaining 0.5% is available for showing up to office hours at least once ever, but ideally in the first full week of class.

The lecturer's email address is `fmanners AT ucsd DOT edu` and his office is AP&M–7343. The TA is Alexander Guldemond; his email is `aguldemo AT ucsd DOT edu` and his office is AP&M–6414.

There is **no required course textbook**. The lecturer will make printed notes available on the course Canvas page. The textbook *Matrix Analysis* (2nd edition) by Horn and Johnson is an optional textbook and may be a useful resource, but we will not follow it closely or refer to it explicitly.

The course calendar below includes details of office hours, as well as the events already mentioned.

### Provisional schedule

The rough, provisional, subject-to-change course schedule is given below.

Item | Date(s) | Description |
---|---|---|

Week 0 | 2021-09-24 | Introduction; recap of linear algebra concepts. |

Week 1 | 2021-09-27 — 2021-10-01 | Inner product spaces; orthonormality and QR. Linear maps and matrices. |

Week 2 | 2021-10-04 — 2021-10-08 | Orthogonal projections; adjoints; self-adjoint, unitary and normal operators. |

Week 3 | 2021-10-11 — 2021-10-15 | Eigenvalues, invariant subspaces, diagonaliziablility. Schur decomposition. |

Week 4 | 2021-10-18 — 2021-10-22 | The spectral theorem for normal operators. Variational problems. |

Week 5 | 2021-10-25 — 2021-10-29 | The singular value decomposition; applications. The polar decomposition. |

Midterm exam |
2021-10-27 | 1400–1450 in AP&M–5402 |

Week 6 | 2021-11-01 — 2021-11-05 | Jordan normal form. |

Week 7 | 2021-11-08 — 2021-11-12 | Norms and metrics. Linear functionals and dual norms; Hahn–Banach. |

Week 8 | 2021-11-15 — 2021-11-19 | Matrix norms. Powers of matrices. Continuity of eigenvalues. |

Week 9 | 2021-11-22 | Angles between subspaces. |

Week 10 | 2021-11-29 — 2021-12-03 | Possible further topics; review. |

Final exam |
2021-12-08 | 1500-1759; location TBD |

### Homework assignments

Those assignments that have not been created yet link to a placeholder.

Week | Deadline | TeX | |
---|---|---|---|

1 | Sunday October 3, 2359 | p1.pdf | p1.tex |

2 | Sunday October 10, 2359 | p2.pdf | p2.tex |

3 | Sunday October 17, 2359 | p3.pdf | p3.tex |

4 | Sunday October 24, 2359 | p4.pdf | p4.tex |

5 | Sunday October 31, 2359 | p5.pdf | p5.tex |

6 | Sunday November 7, 2359 | p6.pdf | p6.tex |

7 | Sunday November 14, 2359 | p7.pdf | p7.tex |

8 | Sunday November 21, 2359 | p8.pdf | p8.tex |

9 | Sunday November 28, 2359 | p9.pdf | p9.tex |

10 | Sunday December 5, 2359 | p10.pdf | p10.tex |