Homepage for MAS212 - Linear Algebra - Spring 2012

Syllabus and course information.

Please come see me if you have any questions.

Office Hours: M-W-F: 13.00 - 14.00.

If these times are not convenient just make an appointment by email or phone!

Office: E6-1 4411

Email: andreash[at]kaist.edu

Phone: 350 7300

Lectures

NOTICES

• May 23:FINAL EXAM see notice below.
• May 04: New Homework. Due date 05/08
• Apr 25: NO CLASS.
• Apr 24: Change of recitation class. Due to KAIST festival we change the recitation classes to Thursday and Friday (with same time and place as other weeks.) Next week back to regular schedule.
• Apr 20: NO CLASS.
• Apr 16: New homework. Due Friday 04/20.
• Apr 11: NO CLASS: Election day.
• Apr 09: CLAIM SESSIONS. (See notice below)
• Apr 08: Grading process complete.
• Mar 28: Exam 2 Wednesday 03/28. (See notice below)
• Mar 16: Exam 1. Friday 03/16: First in Class exam. (See notice below)
• Mar 09: Homework 4. Due Friday 03/09.
• Mar 04: Homework due dat postponed. Due to holiday on Thursday we postpone the due date of Homework 3 to Sunday 03/04.
• Mar 02: NO CLASS. Due to holiday on Thursday there will be no class on Friday 03/02.
• Feb 24: Room change. New room: 3435 Due to College of Science Hooding ceremony.
• Feb 16: Homework 1. Submit in box on 2nd floor of Math Dept.
• Feb 15: Recitation class cancelled. Due to the Graduate Students Welcome Party Wednesdays recitation class is cancelled.
• Feb 14: Recitation classes start
• Feb 10: Room change. New room: 3435 Due to Department of Chemistry Symposium.
• Feb 8: New Lecture room Starting Feb 8, lectures will be in room 1501 in building E6-1 (department of mathematical sciences).

FINAL EXAM

The TAs are currently working on grading the exams and are almost finished.
The schedule will be as follows:

• 05/29 (Tuesday) 11.59. Scores can be downloaded here: scores.
• 05/29 (Tuesday) 8.00 - 10.00 PM Claim Session (room 3430)

Problem set and Solution.

Students will be divided into 2 classrooms based on student id number.

• Room 1501: Student ID numbers from 20070034 to 20110353

• Room 3435: Student ID numbers from 20110365 to 20120993

Notice!

• You will arrive on time to the exam.
• You will bring only writing material to your desk: pencil, pen, eraser.
• You will not bring any of the following items to your desk: papers, books, bags, calculator.
• CELL PHONES will be turned off at all time during the exam.

Results for midterms

The Claim session for the midterms will take place on Monday 04/09: 8 - 10.30 pm. (If you can't make it in this period you must contact the TAs or professor as soon as possible to reschedule.)

Warning: If you make a claim the problem will be entirely regraded: You do stand the chance of getting fewer points!

• Test 1: Room 4411
• Test 2: Room 3430

The grading criteria for TEST 2 can be downloaded here.

Exam 2

Problem set and solution.

Students will be divided into 2 classrooms based on student id number.

• Room 1501: Student ID numbers from 20070034 to 20110353

• Room 3435: Student ID numbers from 20110365 to 20120993

Notice!

• You will arrive on time to the exam.
• You will bring only writing material to your desk: pencil, pen, eraser.
• You will not bring any of the following items to your desk: papers, books, bags, calculator.
• CELL PHONES will be turned off at all time during the exam.

Exam 1

Problem set and solution.

Students will be divided into 2 classrooms based on student id number.

• Room 1501: Student ID numbers from 20070034 to 20110353

• Room 3435: Student ID numbers from 20110365 to 20120993

Notice!

• You will arrive on time to the exam.
• You will bring only writing material to your desk: pencil, pen, eraser.
• You will not bring any of the following items to your desk: papers, books, bags, calculator.
• CELL PHONES will be turned off at all time during the exam.

RECITATION CLASSES AND OFFICE HOURS

There will be two recitation classes. These are not mandatory, but it is highly recommended to attend. For many students, Linear Algebra is the first proper introduction to abstract mathematics, and it contains many new notions and ideas. It can therefore be helpful to attend recitation classes: Don't be afraid to ask questions !!

• Minki Kim
  min901215[at]hanmail.net

  OFFICE HOURS: Monday: 1pm - 2pm and Friday 10am - 12am Room 3427 (E6-1)

• Chinkang Park
  pjiga06[at]kaist.ac.kr
  

  OFFICE HOURS: Tuesdays and Wednesdays after recitation class. Room 3427 (E6-1)

Class 1 :
Room 301, bldg E11
Tuesday, 19.00 -- 20.00

Class 2 :
Room 311, bldg E11
Wednesday, 19.00 -- 20.00

Suggested problems

Below is a list of suggested problems to be covered in recitations. It is recommended to try to solve every problem in the book.

• Chapter 1:
  • Sec 1.2 : Exs 1 , 4 , 5 , 6 , 7 , 8.
  • Sec 1.3 : Exs 2 , 3 , 5 , 6 , 7 , 8.
  • Sec 1.4 : Exs 2 , 4 , 8, 10.
  • Sec 1.5 : Exs 3 , 4 , 5 , 6 , 7 , 8.
  • Sec 1.6 : Exs 4 , 5 , 7 , 9 , 10.
  
  
• Chapter 2:
  Sec 2.1 : Exs 3 , 4 , 6 , 7
  • Sec 2.2 : Exs 2 , 3 , 5 , 7
  • Sec 2.3 : Exs 5 , 6 , 7 , 9 , 13
  • Sec 2.4 : Exs 3 , 4 , 5 , 6 , 7
  • Sec 2.6 : Exs 1 , 2 , 3 , 4 , 5 , 6 , 7
• Chapter 3:
  • Sec 3.1 : Exs 1 , 3 , 4 , 5 , 7 , 9 , 10 , 12 , 13
  • Sec 3.2 : Exs 1 , 3 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12
  • Sec 3.3 : Exs 2 , 3 , 4 , 6 , 7
  • Sec 3.4 : Exs 2 , 3 , 4 , 8 , 9 , 11 , 12
  • Sec 3.5 : Exs 1 , 2 , 3 , 4 , 7 , 8 , 10 , 11
  • Sec 3.6 : Exs 1 , 2
  • Sec 3.7 : Exs 1 , 3 , 4 , 6 , 7
• Chapter 4:
  • Sec 4.2: Exs 1 , 2 , 4 , 5
  • Sec 4.4: Exs 1 , 2 , 4 , 5 , 6
  • Sec 4.5: Exs 1, 2, 5, 8
• Chapter 6:
  • Sec 6.2: Exs 3, 5 , 6 , 7 , 8 , 9 , 10 , 11, 12, 15
  • Sec 6.3: Exs 1 , 2 , 3 , 4 , 5 , 6 , 8, 10
  • Sec 6.4: Exs 1 , 2 , 3 , 4 , 5 , 7 , 8 , 9 , 10 , 11 , 13
  • Sec 6.5: Exs 1, 2, 3
  • Sec 6.6: Exs 1 , 2 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11
  • Sec 6.7: Exs 1 , 3 , 4 , 6 , 8 , 9
  • Sec 6.8: Exs 1 , 2 , 3 , 4 , 5 , 7 , 8 , 9 , 11 , 13
• Chapter 7:
  • Sec 7.1: Exs 1 , 2 , 4 , 6 , 7
  • Sec 7.2: Exs 2 , 3 , 4 , 7 , 11 , 13 , 18
  • Sec 7.3: Exs 1, 2, 3, 4, 5, 6, 8
  • Sec 7.4: Exs 1 , 2 , 4 , 5

Suggested homework

Every week there will be suggested homework . Homework is not mandatory.

Rules about HOMEWORK. If you decide to hand in homework, you WILL hand in your own original solutions. If the TAs suspect that you have COPIED your solution from another student, you will be punished. Students found guilty of handing in copied homework will receive a FAILING grade. The grading of homework is a service that we offer to the students. If students cannot follow the rules we have set, we may choose to stop grading homework. In this case your grade will be determined by the grade on midterm and final alone. Keep this in mind before you hand in copied homework. Not only will you receive a failing grade, but you will ruin the service for everyone else.

You can hand in your homework at Room : 3430 (E6-11) (You can also pick up your graded homework in this room.)

NOTICE: No delay will be accepted!

Homework Claim and Discussion. For students who are interested there will be homework claim and discussion on Mondays 3-5pm in room 3430 led by TA Kim, Minki.

• Homework 1: Due: 02/16 (Thursday) at 23:59:59
  • Section 1.2 : Exercise 8
  • Section 1.5 : Exercise 8
  • Section 1.6 : Exercise 10

• Homework 2: Due: 02/23 (Thursday) at 23:59:59
  • Section 2.1 : Exercise 6
  • Section 2.2 : Exercise 7
  • Section 2.3 : Exercise 7

• Homework 3: Exception, Due: 03/04 (Sunday) at 23:59:59
  • Section 2.4 : Exercise 5
  • Section 2.4 : Exercise 7
  • Section 2.6 : Exercise 7

• Homework 4: Exception, Due: 03/09 (Friday) at 23:59:59
  • Section 3.1 : Exercise 13
  • Section 3.2 : Exercise 12
  • Section 3.4 : Exercise 11

• Homework 5: Exception, Due: 04/20 (Friday) at 23:59:59
  • Section 6.2 : Exercise 15
  • Section 6.3 : Exercise 4
  • Section 6.4 : Exercise 5

• Homework 6: Exception, Due: 05/08 (Tuesday) at 23:59:59
  • Section 6.6 : Exercise 2
  • Section 6.7 : Exercise 3
  • Section 6.8 : Exercise 13

Weekly progress

• 02/06. Problem: Let R be a square of sidelengths 1 and x, where x is a positive irrational number. Show that R cannot be tiled by finitely many squares.

  This is an interesting problem which can be solved by using concepts from Linear Algebra.

• 02/08. Solution to the square tiling problem.

  Overview of the main concepts of Linear Algebra: Fields, Vectorspaces, Linear independence, Linear transformations.

• 02/10. Chapter 1.

  Fields, systems of linear equations, matrices, linear combinations, row equivalence.

• 02/13. Chapter 1.

  Matrices, elementary row operations, invertible matrices. Problem: There are n students that form m student clubs, subject to the following rules: Each club must contain an odd number of members, and every pair of clubs must have an even number of members in common. How large can m be ?

• 02/15. Chapter 2.

  Vector spaces, examples, solution to odd club problem.

• 02/17. Chapter 2.

  Linear combinations, subspaces, linear independence. Problem: There are n students that form m student clubs. Show that if m > n then there are non-empty disjoint subsets I and J of [m] such that the union of the S_i with i in I equals the union of the S_j with j in J.

• 02/20. Chapter 2.

  Basis, dimension, the basis extension property, examples.

• 02/22. Chapter 2.

  Inclusion-exclusion principle for subspaces, coordinates, more on row equivalence, computational questions.

• 02/24. Problem session

  Balanced union: n students, m clubs, m > n.
  Two-distance sets in Euclidean space: How many points can one have in Rⁿ such that only two distances are realized?
  Symmetric matrices ^*: Let S be a symmetric matrix over the binary field where every diagonal element is non-zero. Show that the all 1's vector is in the row space of S.

• 02/27. Chapter 3.

  Linear transformations, null space, range, the rank theorem.

• 02/29. Chapter 3.

  Algebraic properties of linear transformations, the space of linear transformations, linear operators.

• 03/05. Chapter 3.

  Invertible linear operators, isomorphism of vector spaces, matrix representations of linear transformations.

• 03/07. Chapter 3.

  The dual space V^*, the dual basis.

• 03/09. Chapter 3.

  The double dual, the transpose of a linear transformation.

• 03/12. Chapter 4.

  Space of formal power series, subspace of polynomials.

• 03/14. Chapter 4.

  Polynomial division, polynomial ideals, principle ideal generated by p, GCD.

• 03/16. Exam 1.

• 03/19. Chapter 4.

  Irreducible polynomials, primary decomposition of polynomials.

• 03/21. Review of Chapter 3.

• 03/28. Exam 2.

• 04/02. Chapter 6.

  Introduction to canonical forms, diagonalizable operators, characteristic values.

• 04/04. Chapter 6.

  Minimal polynomial of an operator.

• 04/06. Chapter 6.

  Invariant subspaces and conductors.

• 04/09. Chapter 6.

  Necessary and sufficient conditions for triangulable operators.

• 04/13. Chapter 6.

  Necessary and sufficient conditions for diagonalizable operators.

• 04/16. Chapter 6.

  Direct sum decompositions and projections.

• 04/18. Chapter 6.

  Invariant direct sum decompositions.

• 04/23. Chapter 6.

  The Primary decomposition theorem

• 04/27. Chapter 6.

  Decomposition of an operator into Diagonalizable and Nilpotent parts.

• 04/30. Chapter 7.

  Cyclic subspaces. The annihilator.

• 05/02. Chapter 7.

  T-admissible subspaces, the cyclic decomposition theorem.

• 05/04. Chapter 7.

  Cyclic decomposition theorem: proof of existence.

• 05/07. Chapter 7.

  Cyclic decomposition theorem: proof of uniqueness.

• 05/09. Chapter 7.

  Counting similarity classes of matrices.

• 05/11. Chapter 7.

  The Jordan Canonical form

• 05/14. Chapter 7.

  The Smith Normal Form: computing invariant factors.