Most of the reading assignments below refer to the textbook:

[Axler] = Axler, Sheldon J. *Linear Algebra Done Right*. Springer, 2004. ISBN: 9780387982588. [Preview with Google Books]

A few topics not fully covered in the textbook are discussed in supplementary notes available in the Study Materials section. These include:

[F] = Notes on Finite Fields

[OSI] = Notes on One-sided Inverses

[GE] = Notes on Gaussian Elimination

[OB] = Notes on Orthogonal Bases

[ST] = Proof of the Spectral Theorem

[EIG] = Notes on Generalized Eigenvalues

LEC # | TOPICS | READINGS |
---|---|---|

1 | Definition of vector spaces, properties | [Axler] Chapter 1, pp. 2–12. |

2 | Subspaces, sums and direct sums | [Axler] Chapter 1, pp. 13–18. |

3 | Span, independence, bases | [Axler] Chapter 2, pp. 21–31. |

4 | Bases and dimension | [Axler] Chapter 2, pp. 31–34. |

5 | Linear maps, null space / range | [Axler] Chapter 3, pp. 38–47. |

6 | Matrices, invertibility | [Axler] Chapter 3, pp. 48–58. |

7 | Exam 1 on Chapters 1–3 | No Readings |

8 | Finite fields. Systems of equations. | [F], [GE], [OSI] |

9 | Gaussian elimination | [GE] |

10 | Counting over F_{p}. Invariant subspaces. | [Axler] Chapter 5, pp. 75–79, [F] |

11 | Finding eigenvectors | [Axler] Chapter 5, pp. 80–81. |

12 | Upper triangular and diagonal matrices | [Axler] Chapter 5, pp. 81–90. |

13 | Eigenvectors over R and F_{p}
| [Axler] Chapter 5, pp. 91–93, [F] |

14 | Exam 2 on Chapters 1–5 | No Readings |

15 | Inner products, Gram-Schmidt | [Axler] Chapter 6, pp. 97–111, [OB] |

16 | Orthogonal projection, minimization | [Axler] Chapter 6, pp. 111–16. |

17 | Adjoint, self-adjoint, normal | [Axler] Chapter 7, pp. 127–32. |

18 | Spectral theorem | [Axler] Chapter 7, pp. 132–44, [ST] |

19 | Positive operators | [Axler] Chapter 7, pp. 144–47. |

20 | Isometries, polar decomposition | [Axler] Chapter 7, pp. 147–57. |

21 | Exam 3 on Chapters 1–7 | No Readings |

22 | Generalized eigenspaces | [Axler] Chapter 8, pp. 163–68, [EIG] |

23 | Generalized eigenspace decomposition | [Axler] Chapter 8, pp. 173–78, [EIG] |

24 | Characteristic polynomial | [Axler] Chapter 8, pp. 168–73. |

25 | Determinant | [Axler] Chapter 10. |

26 | Trace, canonical commutation relations | [Axler] Chapter 10. |