Call number: | 15106 | |||||||||
Room/Time: | MW 1:10pm--2:25pm, 312 Math | |||||||||
Instructor: | Mikhail Khovanov | |||||||||
Office: | 620 Math | |||||||||
Office hours: | Monday 11-12pm, Wednesday 2:30-4 or email for appointment | |||||||||
E-mail: | ||||||||||
Teaching assistants: |
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Midterm 1: | Wednesday, October 5 | |||||||||
Midterm 2: | Wednesday, November 9 | |||||||||
Final: | Monday, December 19, 1:10pm-4pm, Room MAT 312 | |||||||||
Webpage: | www.math.columbia.edu/~khovanov/linAlgebra2016 | |||||||||
Our teaching assistants hold their office hours in the Help Room in Math 406. The Help Room is open Monday-Thursday 9am-6pm and Friday 9am-4pm. You can go there any time during open hours to get help with the material (not just from our TAs).
Midterm 1 is on Wednesday October 5. It will cover everything up to and including Section 3.1. The bulk of midterm questions will be similar to those on the homework. There will be one multi-part quiz-like
question. To practice for that question and to gain better conceptual understanding of what we've studied so far you can explore problems in the Quiz section at the end of Chapters 1 and 2 of the textbook. Here are some questions to look at, most of them are conceptual and require little or no computation:
Chapter 1, pages 38-40: #1-7, 10, 17, 20, 21, 24, 29, 32, 38.
Chapter 2, pages 107-108: #2-5,7, 10, 17, 19, 21, 27, 28, 34, 39, 43, 50.
Textbook Otto Bretscher Linear algebra with applications, Fifth edition. Cheaper 4th edition is fine too, except for the homework problems, which come from the 5th edition. If you buy the fourth edition, you'll need to get the correct problems from a friend or the library!
Syllabus: Our goal is to cover chapters 1 through 8 of the textbook, with few omissions. The topics are: systems of linear equations and Gaussian elimination, matrices, linear transformations, subspaces, linear spaces, orthogonality and the Gram-Schmidt, determinants, eigenvalues, eigenvectors, symmetric matrices.
Exams: Midterm 1 solutions Midterm 2 solutions
Homework: Homework
will be assigned on Mondays, due Monday the next week before class (excepting
HW1, due Wed Sept 14).
Drop the homework off to the hw box on the 4th floor.
Two lowest homework scores will be dropped.
Graded homework can be
picked up from a tray on the 6th floor (up the stairs, turn right and
through the door, the table with hw trays is immediately to your left).
The numerical grade for the course will be the following linear combination:
15% homework, 25% each midterm, 35% final.
Homework 1, due Wednesday September 14. Read Sections 1.1
and 1.2. Additional Resources section at the bottom of this page
has more examples.
Solve in Section 1.1: #6,10,17, and in Section 1.2: #2,4,6,8. For Section 1.1 problems you can use either elimination or Gaussian elimination.
Solutions of selected problems
Homework 2 due Monday, September 19. Read Sections 1.2
and 1.3.
Solve problems #18,26 from Section 1.2 (pages 18-19) and
problems #2,3,14,18,22,24,28,34 from Section 1.3 (pages 34-35,37).
Solutions of selected problems
Homework 3 due Monday, September 26. Read Sections 2.1,
2.2, and 2.3. Solve problems #6,8,18,20,52 (pages 53-56; in problem 52, compute RREF by hand,
you don't need to use technology) in Section 2.1,
#8,20,26abcd (pages 71-72) in Section 2.2, and #2,4,10,34,56 (pages 85-86) in Section 2.3.
Solutions of selected problems
Optional problems, for additional practice (do not turn in):
Section 2.1 #22,28,30; Section 2.2 #19,21,28,30; Section 2.3 #8,14,43.
Homework 4 due Monday, October 3. Read Sections 2.4,
3.1. Solve problems #4,8,12,16,32,38 in Section 2.4
(pages 97-98) and #14,16,18,30 in Section 3.1 (pages 119-120).
Solutions of selected problems
Homework 5 due Monday, October 10. Solve problems #2, 6, 8, 38, 40, 49 in Section
3.1
Solutions of selected problems
Homework 6 due Monday, October 17. Solve problems #18,22,24,32,38 in Section
3.2 (in problem 18 only identify redundant vectors). Solve problems
#8,14,16,22,26a,30 in Section 3.3.
Solutions of selected problems
Homework 7 due Monday, October 24. Solve problems #12, 24, 26, 36, 42, 44 in
Section 3.4, problems #12, 14, 20, 22, 28 in Section 4.1, and
problems #6, 14 in Section 4.2. For problem 24 in Section 3.4, solving
it in just one way (rather than in 3 different ways, as the question asks) is enough.
Solutions of selected problems
Homework 8 due Monday, October 31. Solve problems #6, 12, 18, 22, 34, 42abc, 60abc in
section 4.3 and problems #6, 10, 28 in Section 5.1.
Solutions of selected problems
Homework 9 due Wednesday, November 9. Solve problems #2, 4, 14, 32, 38 in
Section 5.2 (in problems #2, 4, 14 also determine the QR-factorization -- these are problems 16, 18, 28 in the same section). Solve problems #4, 36, 54, 69 in Section 5.3. The book has an answer to problem #69, so please show your working leading to the correct answer. For additional practice and to become comfortable with symmetric, skewsymmetric, and orthogonal matrices, I recommend that you do a random selection of problems among #5-26 in this section, but do not write solutions to them in your homework.
Solutions of selected problems
Homework 10 due Monday, November 21.
Read Sections 6.1 and 6.2.
Solve problems #10, 16, 38, 40, 42, 54 in
Section 6.1 and problems #5, 6, 16, 18, 40, 46 in Section 6.2.
Solutions of selected problems
Homework 11 due Monday, November 28.
Read Section 6.3 and solve problems #3, 4, 7, 8(explain why the equality still holds if the vectors are not independent), 10 (Hint: how does the length change when you convert a vector to one orthogonal to a subspace?), 13, 22.
Solutions of selected problems
Homework 12 due Monday, December 5.
Read Sections 7.1-7.3 and solve problem 30 (Section 6.3), problems
6, 12, 18, 56 (Section 7.1), problems 4, 10, 18, 22 (Section 7.2), problems
2, 4, 40 (Section 7.3).
Homework 13 Do not turn in! Practice problems to cover the material of last week of lectures, Sections 7.3-7.5. Solve problems
8,9 in Section 7.3, problems 1, 6, 13 in Section 7.4 and problems
11, 20, 21, 26, 28 in Section 7.5.
Dates | Topics | Book |
Wed 09/07 | Linear equations, matrices, vectors, GJ elimination | 1.1, 1.2 |
MW 09/12, 09/14 | RREF, Solutions of lin. systems, linear transformations | 1.2, 1.3, 2.1 |
MW 09/19, 09/21 | LT in geometry, products, inverses | 2.2, 2.3, 2.4 |
MW 09/26, 09/28 | Image, Kernel of LT, subspaces, bases, independence | 3.1, 3.2 |
Mon 10/3 | Review | 3.2 and Review |
Wed 10/5 | First Midterm | 1.1-3.2 |
MW 10/10, 10/12 | Dimension of subspace, Coordinates | 3.3, 3.4 |
MW 10/17, 10/19 | Linear spaces, linear transformation | 4.1,4.2 |
MW 10/24, 10/26 | Matrix of linear transformation, orthonormal bases | 4.3, 5.1 |
MW 10/31, 11/2 | Gram-Schmidt orgonalization, Orthogonal matrices, Inner products, | 5.2, 5.3, 5.5 |
Mon 11/7 | NO CLASS (Academic Holiday) | |
Wed 11/9 | Second Midterm | |
MW 11/14, 11/16 | Determinants | 6.1, 6.2 |
MW 11/21, 11/23 | Determinants, Cramer's rule, Thanksgiving | 6.3 |
MW 11/28, 11/30 | Eigenvalues and eigenvectors | 7.1-7.3 |
MW 12/5, 12/7 | Complex eigenvalues, Jordan normal form | 7.5 |
Mon 12/12 | REVIEW | 1.1-7.5 |
TBA | FINAL EXAM |
Examples of Gaussian Elimination
A quick introduction to Row Reduction (for now, read sections 1 and 2 only).