**Meeting time:**MWF 10-11, Room 4-237**Final Exam: Thursday, December 19, 9-12, Room 50-340**(Walker Memorial, near the corner of Ames Street and Memorial Drive).**Text:**Sheldon Axler,*Linear Algebra Done Right*, third edition**Text videos**by author (and former MIT instructor) Sheldon Axler. These are more less exactly the text, presented as nice slides and read aloud. Closed captions are available if they're helpful for you, including automatic translations into a wide variety of languages. (The captions are not great, particularly for the formulas, but they're good enough to follow the slides without the sound.)**Syllabus:**pdf file for syllabus (modified 11/25 to change due date for PS9 to**Wednesday December 4**.).- I've used the notes below when teaching this class in other years. Don't know how much I'll use this year.
**Some interesting bases:****Notes on finite fields:**for lectures 9/30, 10/7...**Notes on one-sided inverses:**talked about this material in class 9/25, but I won't use the notes explicitly.**Notes on Gaussian elimination:**for lectures 10/2 and 10/4. Cleaned up slightly 10/1/19. The only serious change from the old version (dated 2013) is correcting a typo on page 18: the lower right entry of the last displayed matrix should have been 1, not 0.**Example from class October 2**with the arithmetic corrected. This also serves as an example of reduction to reduced row-echelon form as described in the preceding file.**Notes on computing eigenvalues:**for lectures 10/9 and 10/11.**Notes on orthogonal bases:**for lecture 10/30.**Proof of the Spectral Theorem:**for lecture 11/6.**Notes on matrix formulations of various ideas about linear maps :**intended in part as a review for the first two-thirds of the class.**Proof of polar decomposition :**promised in lecture 11/22.**Notes on generalized eigenvalues :**for lectures after 11/22. Updated 11/18 to refer to _third_ edition of Axler. Version generalizedB added 11/20 has some tiny cosmetic corrections.**Mathematical formalism of quantum mechanics:****Practice questions for the final exam.***Caveat emptor*: see the detailed disclaimers in the document.

Solutions.

Update 12/9: the previous version gave a solution to a more difficult question than was on the practice questions. (Thank you for pointing this out, Agnes!) This version includes both the original easy question and the more difficult one, and solutions to both.

Update 12/12: Solution to 1(e) was based on a definition I gave in class, which was different from the definition in the text and the rest of the world. (I need hardly add that the rest of the world is wrong.) (Thank you for sending me a message from the world, Evan!) Solutions (version C, linked above) now acknowledge this difference. I'll try to avoid questions on the real final depending on differences like this.

Update 12/14: there was a sign error in 5(d) (thank you Agnes!). Corrected in the current links above (with a D in the file names; possibly an ill omen, but we Don't Do omens, Do we?)

**Subject evaluations** are open from **now until 9:00 a.m. Monday December 16**. These evaluations are taken very seriously both by your fellow students and by the math department; comments can have an important effect on how subjects will be taught in the future. Please participate!

**Instructor**: David Vogan, 2-355, dav@math.mit.edu, 617-253-4991

**Office hours**: Thursday 3-4:00, Friday 4-5 in 2-355, including 12/5-6.

**Office hours for final exam:** Thursday December 12 3-5, Friday December 13 4-5:30, Monday-Wednesday December 16-18 2-4 p.m.

I was asked to advertise a concert that's raising funds for the Dana Farber Cancer Institute in Boston, orginally scheduled for Monday, December 9. The event is **POSTPONED** to spring 2020. I'll leave the other info up, because why not. Here's a link to the musician. I can certainly attest that Dana Farber is a great institution to support, and I've verified that Dana Farber thinks this is a fundraising event for them. I listened to some video of the singer; she sounded very talented, but not too much to my taste. Probably you should take that as a strong recommendation. (I loved the guitarist; but probably you should take that as a negative.)

Problem sets will be posted at least a week before they are due. Solutions will be posted the afternoon after the problem sets are due.

pdf file for Problem Set 1

pdf file for solutions to Problem Set 1 (Partly in fairness to others using this text, I will **not** post solutions to the problems assigned from the text.)

pdf file for Problem Set 2

pdf file for solutions to Problem Set 2

pdf file for Problem Set 3

pdf file for solutions to Probl em Set 3

pdf file for Problem Set 4

pdf file for solutions to Problem Set 4

pdf file for Problem Set 5

pdf file for solutions to Problem Set 5

pdf file for Problem Set 6

pdf file for solutions to Problem Set 6.
CORRECTED 11/7/19: in 2(b), the fourth entry in Vc was calculated incorrectly.

pdf file for Problem Set 7

pdf file for solutions to Problem Set 7

pdf file for Problem Set 8

pdf file for solutions to Problem Set 8 CORRECTED 12/3/19 thanks to Quinn Brodsky. (Wrote A instead of A^{1/2} several times; wrote lambda instead of lambda^{1/2}.)

pdf file for Problem Set 9.

pdf file for solutions to Problem Set 9

Here's a distribution of scores on the final, and what letter grade each range corresponds to. (Of course getting an X on the final does not guarantee getting an X for the class.)

A 185-200: 26

B 170-184: 12

C 150-169: 13

D 140-149: 2

below 140: 3

Parallel for semester totals:

A 600-707: 33

B 500-599: 17

C 400-499:: 6

As I've explained in class, the grade you received is not taken directly from your point total for the semester. The list shows the number of As in the class, not the (smaller) number of people with the indicated point totals.

After I posted a score distribution on Exam 3 to help you understand what happened in a situation where a great many people should be disturbed about their scores, several people asked for score distributions on all three exams.

range | Exam 1 | Exam 2 | Exam 3 |
---|---|---|---|

90-100 | 26 | 20 | 6 |

80-89 | 13 | 15 | 3 |

70-79 | 10 | 13 | 13 |

60-69 | 5 | 6 | 11 |

50-59 | 6 | 6 | 9 |

40-49 | 1 | 2 | 13 |

0-39 | 0 | 0 | 5 |

average | 83.5 | 81.8 | 61.7 |

std dev | 15 | 15 | 18 |

Here's how Exam 3 looked to me. There was a good bit of content covered in class and in the notes but not so much in the text. Many people had a lot of difficulty with that material. There were some questions asking you to reproduce proofs of simple facts about self-adjoint operators. Those also caused a lot of grief, perhaps in part because I have not often asked such questions in the past.

Despite these "dangerous bends," the exam was entirely material that I (still) think it's reasonable for you to understand. On the other exams, I've said that the class average might be thought of as approximately a low B. On this exam I don't say that: I think that a reasonable lower bound for a B-level understanding is more like 75.

I've also said that I'm interested more in where your understanding ends up than in how you get there. If your work on the final shows a good understanding of the material you didn't get on this exam, I'll be happy.