Section 003 (11:15am): in MUEN E118, from 1:30pm until 4pm, on 11 December
Section 005 (1:25pm): in ECCR 151, from 4:30pm until 7pm, on 13 December
If A is a Markov matrix (we have thought of this as a matrix of probabilities of clicking on links on different websites) and x is a vector then we think of Ax as the vector obtained by redistributing the entries according to the probabilities in the matrix A. Can you think of an interpretation of what AT does? Let me know what problems you would like to review. No submission is required for Wednesday.
In class, we conjectured that the dimension of the 1-eigenspace of a Markov matrix A is equal to the number of connected components of the graph representation of A. Test this conjecture on at least two Markov matrices that we haven't considered in class. Do §6.3, #11, 13 and §10.3, #10. Post your answers on Canvas. Let me know what you want to review.
Problem Set #3: due on Friday, 2 December. Submit solutions to these problems on Canvas.
Read §6.3 and watch Strang's Lecture 23. You won't be responsible for difference equations or stability, so you may skip those parts. You may also want to review complex numbers, from §9.1 and/or 3blue1brown. Do §10.3, #3 (try to do these with as little calculation as possible), 4, 9 and §6.3, #10, 18, 19, 26. Post your answers on Canvas. Ask questions.
Exam #3 Revisions: meet to discuss the exam before 14 November; select problems by 18 November; submit completed problems by 28 November on Canvas. Suggested problems (may be updated).
Watch Chapter 13 of Essence of Linear Algebra. Do §6.1, #33, §6.2, #1, 2 (don't ignore the XΛX-1 this time), 3, 4, 5, 7, 9, 25. Submit your work on Canvas. Ask questions.
Do §6.1, #30, 32. Complete the problems on the handout. Read §6.2 and watch Strang's Lecture 22. (Spoiler alert: §6.2 contains answers to the problems on the handout, but with different notation. Warning: the formula Strang writes at about 32:30 is incorrect — what should he have written?) Submit your work on Canvas. Ask questions.
Watch the eigenvector video on 3blue1brown. Do §6.1, #29, 8, 9, 11, 19, 21, 24, 26, 27 (some of these are repeated from last time; if you felt like you understood them then, feel free to skip them now) and §6.2, #2 (calculate the way we did in class — ignore the XΛX-1). Post your solutions on Canvas. Ask questions.
(due Wednesday, 9 November) Read §6.1 and watch Strang's Lecture 21. Do §6.1, #1, 2, 3, 4, 5, 8, 9, 27. Post your answers on Canvas. Ask questions.
(due Wednesday, 2 November) Do §5.1, #22, 23 and §5.2, #15 and §5.3, #41 (intentionally repeated; hint: do row expansion using A and column expansion using B). Post your answers on Canvas.
Problem Set #2 Revisions (optional) are due on Monday, 31 October. You can submit your revision on Canvas after you have discussed your problems with me.
(due Monday, 31 October) Do §5.2, #13, 16, 19, 23, 27, 33 and §5.3, #7, 8, 9, 10, 39, 41. Post solutions on Canvas. Required: ask a question.
(due Friday, 28 October) Read §5.3 and watch Strang's Lecture 20 (we did not finish discussion of these on Wednesday). Do §5.3, #4, 16, 18, 19, 21, 23, 26. Post your answers on Canvas. Required: ask questions.
(due Wednesday, 26 October) Finish the first page of Handout #2. Read §5.2 and watch Strang's Lecture 19. Do §5.2, #1, 2, 11. Post your work on Canvas. Required: ask questions.
(due Monday, 24 October) Do §5.1, #11, 12, 15, 17, 18 (I suggest using column operations rather than row operations), 29. Post your solutions on Canvas. Watch Chapter 6 of Essence of Linear Algebra (you may wish to refer back to some of the earlier videos if you want to understand the discussion of equally spaced lines at the beginning, but it won't be necessary to understand that part for Monday). Required: ask questions.
Exam #2 revisions will follow a different process from previous revisions. For Monday, 24 October, please submit up to 2 problems from the updated Exam 2 problems, along with your original Exam 2. Your work on these problems will replace your work on the original exam and your grade will be updated accordingly. Please follow the same rules for collaboration as on the problem sets: feel free to discuss the problems with other memebers of the class, but write up your final submission without consulting any sources (including notes).
(due Friday, 21 October) Read §5.1 and watch Strang's Lecture 18. Do §5.1, #1, 3, 9, 13. Post your answers on Canvas. Required: ask questions.
(due Wednesday, 19 October) The problems from Exam #2 will be posted on Tuesday. Please look over them: we will discuss them in class on Wednesday. Nothing needs to be submitted on Canvas.
Exam #2: Monday, 17 October focussing on Chapter 4, Orthogonality. [exam]
(due Friday, 14 October) Do §4.3, #28 and §4.4, #1, 3, 5, 8, 10, 13, 14, 18. Post your answers on Canvas. Optional: make suggestions for next time.
(due Wednesday, 12 October) Read §4.4. Do §4.2, #4, 7 (the projection matrix onto C(A) is A(ATA)-1AT — why does this matrix do the same as the process we described in class?), 13, 17, 19, 20, 23, 27 and §4.3, #1, 5, 7, 9. Post your solutions on Canvas. Optional: make suggestions for next time.
(due Monday, 10 October) Read §4.3. Do §4.1, #6, 11, 16, 17, 24, 30 and §4.2, #2. Post your answers on Canvas. Optional: make suggestions for Monday's class.
Problem Set #2 is due on Friday, 7 October. Complete these problems and submit your solutions on Canvas. You will probably want to make use of the matrix calculator (you may want to revisit this page to recall the commands). Don't forget to include a references page. [solutions]
(due Wednesday, 5 October) Read §4.2. Do §3.5, #11, 23, 26 and §4.1, #3, 4, 13, 14, 15, 18, 19, 22, 26. Post your work on Canvas. Optional: make suggestions for next time.
(due Monday, 3 October) Read §4.1. Do §3.4, #14, 15, 16, 20, 21, 23, 43 and §3.5, #6, 11, 14, 24, 31. Because of my mistake assigning problems for Friday, you may have already attempted some of these problems. If you don't think you will get anything out of looking at the problems again, there is no need to do any problems that you have already attempted. Submit your work on Canvas. Optional: make suggestions for Monday's class.
Exam #1 revisions are due on Monday, 3 October. Here are some suggested problems (will be updated as people suggest more problems).
(due Friday, 30 September) Read §3.5; do §3.3, #28, 31, 33 and §3.4, #2, 6 (U is supposed to be obtained from A by row operations), 12, 14, 15, 16, 20. Post your answers to Canvas. Optional: make suggestions for Friday.
(due Wednesday, 28 September) Read §3.4; do §3.3, #1, 2, 6 (also write the column space as the null space of another matrix), 10, 11, 12, 15 (U is upper triangular), 22, 24 and §3.4, #1, 11. Post your answers on Canvas. Optional: make suggestions for Wednesday.
(due Monday, 26 September) Read §3.4; do §3.2, #2, 10, 13, 14, 21, 32 (can you find a way to answer this question from the image, without doing any calculation?), 33, 37 and §3.3, #5. Submit your work on Canvas. Optional: make suggestions for the next class.
(due Friday, 23 September) Read §3.3; do §3.1, #10, 16, 22, 23, 24, 25, 29 and §3.2, #5, 20, 22. Submit your solutions on Canvas. Optional: give feedback on the exam and make suggestions for next class.
(due Wednesday, 21 September) Read §§3.1, 3.2; do §3.1, #4, 6, 15ab, 19, 22. These problems should be a warmup for the discussion of vector spaces on Wednesday. We won't go over any of these problems in class.
Problem Set #1 revisions: Wednesday, 21 September. If you are planning to do revisions, remember to discuss the topics and problems with me beforehand. You may wish to use some of these problems, some of which were suggested by your classmates.