MATH 235 - Linear Algebra 2
(fall semester 2002)

• Final/Midterm/Assignment Marks for Section 1 available now.
• Text: The reference text Elementary Linear Algebra authors: B. Kolman and D. Hill is now available on 3 hour reserve; CALL NUMBER IS UWD1464.
• Directions to students:
  "Feel free to discuss the assignments with your colleagues, but write the final solutions on your own, and acknowledge those who contributed ideas for your solutions." See help on avoiding plagiarism

Links to:

• Midterms/Final/Grades
• Assignments
  • Assignments Sections 01,05
  • Assignments Sections 02,03,04
• Solutions
• Tutorial Center - MC 4067
• More Interesting Links Below
  • Eigenvalues/Eigenvectors:
    • Current Headline News - World's Largest Matrix Computation;
      Google's PageRank is an eigenvector of a matrix of order 2.7 billion.
    • Population Dynamics:
      See the matlab file popdyn.m, to illustrate transitions of a Markov chain in population dynamics; convergence to a steady state.
    • One way to solve an IVP is to use the exponential of a matrix. See the related paper: "Nineteen dubious ways to compute the exponential of a matrix." in SIAM Review Volume 20 (1978), no. 4, pages 801--836. ( MathSciNet abstract)

Math235 is the second of two terms of linear algebra. We examine linear transformations acting on vector spaces, together with their matrix representations. Determinants are introduced and empoloyed primarily in the evaluation of eigenpairs. Diagonalization and orthogonal diagonalization are explained, and their importance is revealed through various applications.
Prerequisite: M136 or equivalent - it is assumed that the student is familiar with Complex Inner Product Spaces.
Textbook: Math 235 Course Notes by Conrad Hewitt, available from the Math Copy Center - MC 5182.
The Course Notes are available online - including typos and sample exams. An excellent reference is the text Elementary LInear Algebra authors: B. Kolman and D. Hill also available on 3 hour reserve CALL NUMBER IS UWD1464.

S. Lectures Room   Instructor    Office       Email               Office-hours
01 12:30MWF RCH302 H Wolkowicz   MC6065 hwolkowicz@uwaterloo.ca   M2:30-3:30,T10-11
02 11:30MWF RCH302 I VanderBurgh MC5098 iwtvanderburgh@uwaterloo.ca  ???????
03 08:30MWF RCH301 P Crippin     MC5099 pwcrippi@uwaterloo.ca    ???????
04 08:30MWF RCH302 I VanderBurgh MC5098 iwtvanderburgh@uwaterloo.ca  ???????
05 09:30MWF STJ2009 C Hewitt     STJ106 cghewitt@uwaterloo.ca     ???????

Information related to the teaching activities of Professor Henry Wolkowicz; The material is related to the course taught in the FALL SEMESTER 2002. DISCLAIMER: Due to the nature of WWW, links and information get out of date. I provide these links in the hope that they help with the course as well as create additional interest. I do try and keep these links as current as possible, i.e. links may be added and/or deleted during the semester.
PLEASE

• Directions to students:
  "Feel free to discuss the assignments with your colleagues, but write the final solutions on your own, and acknowledge those who contributed ideas for your solutions."
• Assignments:10%, Midterms: 15% each, Final: 60%.
• The second midterm is on: Tuesday 4:30-6:30, Nov. 5th.
  The midterm rooms for the 5 sections are, respectively:
  MC1056/85, DC1350, MC 4059/61, DC 1351, MC2017.
  The double degree students start at 6PM and write in MC2035.
• final exam scheduling:
  MATH 235, 13-Dec-02, Friday morning, 9:00-12:00.

  To be announced:
  Alphabetical    in rooms in MC    
  A--             in 1085 4063 
  A--         
  A--             4020        
  A--             4041, 4042, 4045, 4058, 4059, 4060 and 4061
  

• Old midterms/finals are available at the MathSoc Online Exambank. Be aware that previous midterms covered more material, since there was only one midterm during the semester.
• Midterm/Assignment Marks for Section 1 available now.

Section 01 uses math drop boxes outside of MC 4066
(by 12 (noon) for H.W.) 

Section 01:   box #3
Slot#:         1    2    3    4 
Division:     A-F   G-L  M-R  S-Z
(One assignment will not be counted in the final grade.)

1. Assignment
   • is due Monday Sept. 16. Do problems 3,4,9 from the Winter 2002 Math 136 final exam. (The exam can be found near the beginning of the Course Notes)
   • Comments from Markers.
2. Assignment from Problem Set #1
   is due Wednsday Sept. 18. Do problems 1,4,5,6 from Problem Set #1 in the Course Notes (Page 11).
   Comments from Markers.
3. Assignment from Problem Set #2
   is due Wednsday Sept. 25. Do problems 2,3,6,8 from Problem Set #2 in the Course Notes (Pages 18-19).
   (From PS3: Problem 1 (only do number 2 from PS#2), and Problems 2,5,7 (Page 27). - for Section 05 ONLY)
4. Assignment from Problem Set #3 and #4 and #5
   is due Wednsday Oct. 2.
   
   (From PS3: Problem 1 (with only number 2 from PS#2), and Problems 2,5,7 (Page 27). - for Section 01 ONLY)
   From PS4: Problems 1,3,4a), 4b)(iv) (Page 32). (for BOTH sections)
   (PS5 1i(b), 1ii, 2,3 - - for Section 05 ONLY)
   Comments from Markers.
5. Assignment from Problem Set #5
   is due Wednsday Oct. 9.
   From PS5: 1i(b), 1ii, 2,3 - - for Section 01 ONLY
6. Assignment from Problem Set #6
   is due Wednsday Oct. 16.
   From PS6: 1a, 2bi, 3a, 5,6 - - for Section 01 and 05
7. Assignment from Problem Sets #7,8
   is due Wednsday Oct. 23.
   From PS7 1, 3, 4, 6 (do not hand in 1)
   From PS8 1, 3, 4 (do not hand in 3)
   --for Section 01 and 05
   Comments from Markers.
8. Assignment from Problem Sets #9,10
   is due Wednsday Oct. 30.
   From PS9 probs 6,7
   From PS10 probs 2,3,4 (do not hand in 4)
   --for Section 01 and 05
9. Assignment from Problem Sets #11
   The following questions from PS11 are due Monday Nov 4 (the day before the midterm).
   From PS11 do only probs 1a and 4, (do probs 2a)b)c) and 6 but do not hand them in )
   --for Section 01 and 05
10. Assignment from Problem Sets #12
    is due Wednsday Nov. 13.
    From PS12 probs 1,2,3,5,6, (do not hand in 1)
    --for Section 01 and 05
    And for Section 01, in addition, find the eigenvalues and eigenvectors (diagonalize if possible) the three matrices given in class - but do not hand in.
    Comments from Markers.
11. Assignment from Problem Sets #13#14
    is due Wednsday Nov. 20.
    PS 13: 1b),1c),1g),2,3,5 (do not hand in 1b) and 2 and 3)
    PS 14: 1,2,6a),6b),6e),6f),6g) (do not hand in 2 and 6f) and 6g) )
    --for Section 01 and 05
    
12. Assignment from Problem Sets #15#16
    is due Wednsday Nov. 27.
    PS 15: 1a), 3, 6b)
    (Note: T_0 is the 0 operator; T_I is the identity operator. See also Q3 from PS12. Also NOTE that PS 15 #3 and PS 16 #2 were not checked for correctness by the markers due to time pressure. Please check the solutions for these.)
    PS 16: 1, 2
    And Exercises 1 and 2 from: notes and exercises on the Minimum Polynomial.
    --for Section 01 and 05
    
13. Assignment from Problem Sets #17#18#19
    Do NOT hand in!
    PS 17: 1,2,3
    PS 18: 6,8
    PS 19: 2
    --for Section 01
    But: PS 18: 8 HAND IN --for Section 05
    

1. Assignment
   is due Monday Sept. 16. Do problems 3,4,9 from the Winter 2002 Math 136 final exam. (The exam can be found near the beginning of the Course Notes)
2. Assignment from Problem Sets #1,#2
   
   PS1 1c, 1d, 2b, 4, 5, 6
   PS2 2, 3, 5, 6, 7, 8
3. Assignment from Problem Sets #3,#4:
   is due Wednsday Oct. 2.
   Problem Set #3: Problems 1.ii), 1.iii), 2, 4, 7
   Problem Set #4: Problems 1;3;
4. Assignment from Problem Sets #6:
   
   is due Wednsday Oct. 16.
   Problem Set #6: Problems Hand in 1b,2b,5
   Problem Set #6: Problems Don't Hand in 1a,3a
5. Assignment from Problem Sets #7,8
   is due Wednsday Oct. 30.
   From PS7 1, 3, 4,
   From PS8 1, 3, 4
   --for Sections 02,03,04
6. Assignment from Problem Sets #12,13
   is due Wednsday Nov. 13.
   From PS 12: to hand in: 1, 2; not to hand in: 3, 5
   From PS 13: to hand in: 1b, 1c, 2, 3; not to hand in: 5
   --for Sections 02,03,04
7. Assignment from Problem Sets #15,16
   is due Wednsday Nov. 27.
   PS15 1a, 1c, 4 (don't hand in 1a)
   PS16 3
   MinPoly: And Exercises 1 and 2 from: notes and exercises on the Minimum Polynomial.
   --for Sections 02,03,04

Tutorial Center - MC 4067

Interesting Links

• Math 235 web page, Fall/01
• Summer Undergraduate Research Program, Undergraduate Research in C&O at Waterloo.
• Essay from Gilbert Strang on importance of Linear Algebra. (Link to MIT, linear algebra with VIDEOS!!!)
• Math Forum at Drexel. With discussion groups, Ask Dr. Math, Problems of the Week, ... e.g. see Q&A College Level and Beyond; linear Algebra archives
• History of Matrices and determinants. (Sylvester originated the term matrix.)
• On the Centrality of Linear Algebra in the Curriculum, by Carl C. Cowen, at The Mathematical Association of America.
• Further exercises in matlab can be obtained from the ATLAST Project at the Library of ATLAST M-files
  Also: VLA - Visual Linear Algebra, which runs using matlab on PCs.
• Eigenvalues, including information on the minimum polynomial, at The Matrix Reference Manual
• The Cayley-Hamilton Theorem, mentioned in two excercises in the text; Also some historical papers and a simple proof by Dr. Math.
• Student Projects in Linear Algebra. (See in addition: Linear Algebra Gems: Assets for the Undergraduate Linear Algebra Curriculum, Mathematical Association of America (2002), 328 pages (with D. Carlson, C. Johnson, D. Lay) and Resources for Teaching Linear Algebra, edited by David Carlson, Charles R. Johnson, David C. Lay, A. Duane Porter, Ann Watkins, and William Watkins. MAA Notes, vol. 42. MAA, 1997. ISBN 0-88385-150-4)

H. Wolkowicz Home Page