## Math 202A (Fall 2019)

### 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–5402.
Note there is *no* lecture on Wednesday November 27 or Friday November 29 to allow for Thanksgiving.

There will be a **midterm exam** on Wednesday 30th October, during usual class time. There is a **final exam** on Wednesday December 11, 1500—1759.

**Homework** will be set every week, due by the start of class on Fridays, starting at the end of week 1 (so, 1400 on Friday October 4); the exception is Thanskgiving week (see below). Work should be handed in via Gradescope.

The **grade breakdown** is 20% for the midterm, 30% for homework and 50% for the final exam.

The lecturer's email address is `fmanners AT ucsd DOT edu` and his office is AP&M–7343. The TA is Ziyan Zhu; his email is `ziz276 AT ucsd DOT edu` and his office is AP&M–1121.

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 | 2019-09-27 | Introduction; recap of linear algebra concepts. |

Week 1 | 2019-09-30 — 2019-10-04 | Inner product spaces; orthonormality and QR. Linear maps and matrices. |

Week 2 | 2019-10-07 — 2019-10-11 | Orthogonal projections; adjoints; self-adjoint, unitary and normal operators. |

Week 3 | 2019-10-14 — 2019-10-18 | Eigenvalues, invariant subspaces, diagonaliziablility. Schur decomposition. |

Week 4 | 2019-10-21 — 2019-10-25 | The spectral theorem for normal operators. Variational problems. |

Week 5 | 2019-10-28 — 2019-11-01 | The singular value decomposition; applications. The polar decomposition. |

Midterm exam |
2019-10-30 | 1400–1450 in AP&M–5402 |

Week 6 | 2019-11-04 — 2019-11-08 | Jordan normal form. |

Week 7 | 2019-11-11 — 2019-11-15 | Norms and metrics. Linear functionals and dual norms; Hahn–Banach. |

Week 8 | 2019-11-18 — 2019-11-22 | Matrix norms. Powers of matrices. Continuity of eigenvalues. |

Week 9 | 2019-11-25 | Angles between subspaces. |

Week 10 | 2019-12–02 — 2019-12–06 | Possible further topics; review. |

Final exam |
2019-12–11 | 1500–1759; location TBD |

### Homework assignments

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

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

1 | Friday October 4, 1400 | p1.pdf | p1.tex |

2 | Friday October 11, 1400 | p2.pdf | p2.tex |

3 | Friday October 18, 1400 | p3.pdf | p3.tex |

4 | Friday October 25, 1400 | p4.pdf | p4.tex |

5 | Friday November 1, 1400 | p5.pdf | p5.tex |

6 | Friday November 8, 1400 | p6.pdf | p6.tex |

7 | Friday November 15, 1400 | p7.pdf | p7.tex |

8 | Friday November 22, 1400 | p8.pdf | p8.tex |

9 | Monday December 2nd, 1400 | p9.pdf | p9.tex |

10 | Friday December 6, 1400 | p10.pdf | p10.tex |