Math 54 -- Linear Algebra and Differential Equations
Fall 2003, Prof. Allen Knutson

  • Tuesday/Thursday, 11 AM - 12:30 PM, in 2050 Valley Life Sciences
  • Books:
  • Hill, 3rd Edition, "Elementary Linear Algebra"
  • Boyce & DiPrima, 7th Edition, "Elementary Differential Equations and Boundary Value Problems"
  • Worksheets at Copy Central, on Bancroft
  • Tests (including the final): one double-sided page of handwritten notes, no calculators.
  • Two in-class midterms: October 2, November 6 (both Thursday)
  • Review sessions for the final :
  • Zach, 141 McCone, Dec 4, 5-8 PM
  • Ben, Tuesday, 1-3 PM, 10 Evans
  • Marco, Tuesday, 4 PM, 81 Evans
  • Peter, Monday night 6-8 in 9 Lewis
  • Yuan, Tuesday night 6-8 in 9 Lewis I THINK
  • Answers to midterm #1 here
  • Answers to midterm #2 here
  • Final exam group 9, Friday December 12, 5-8pm, in 100 Haas Pavilion
  • Homework: assigned Tuesday in class, due Monday in section
  • Quizzes: Wednesdays in section
  • Grading (more detail here):
  • Quizzes 20%
  • 1st midterm 20%
  • 2nd midterm 20%
  • Final 40%
  • Prof. Knutson's office hours: Wednesday 9-11 AM, Thursday 12:40-2, in 1033 Evans
  • This page: http://math.berkeley.edu/~allenk/courses/fall03/54/

    • This class is very, very full. Note that there is still some room in Math 54M.

    For enrollment issues, you must see the Head TA, Kendra Eyer, in 1070 Evans, and fill out a processing form. If you do not fill out that form, you cannot be added into the class. Conversely, if you are already enrolled in the class, you do not need to fill out this form! Her office hours are Tue 1-4, Wed 4-5, Thu 1-4, Fri 2:30-5.

    TAs (all of these emails are @math.berkeley.edu):
    Name Section 1 Section 2 Office hours email
    Alice Medvedev11 AM, 87 Evans10 AM, 4 Evans M 12-1,2-3 741->712 Evansalice@math
    Ari Nieh 8 AM, 3 Evans 10 AM, 425 Latimer M 11-12, W 3-4, 710 Evans ari@math
    Darren Rhea Noon, 87 Evans3 PM, 45 Evans M,T 4-5, 1070 Evansdrhea@math
    Ben Hough 2 PM, 51 Evans3 PM, 41 Evans M 4-5, W 1-2jbhough@math
    Marco Zambon8 AM, 87 Evans9 AM, 103 Moffitt T 1:40-2:40, F 2-3, 1095 Evanszambon@math
    Peter Tingley Noon, 6 Evans1 PM, 7 Evans M 10:30-11:30, F 2-3pwtingle@math
    Yuan Yao 10 AM, 237 Cory1 PM, 9 Evans M 2-3, Th 4:30-5:30, 759 Evans yao@math
    Zachary Judson11 AM, 3111 Etcheverry2 PM, 39 Evans M 3-4, W 12-1 in 814 Evansjudson@math
    Paulo Ney de Souza10 AM, B51 Hildebrand12 noon, 3109 Etcheverry MF 11-12, 999->1087 Evansdesouza@math
    (You'll have to type the address in yourself -- if they were clickable, then web-searching robots would sign them up for spam.)

    What we covered, and what could be on the final

  • Hill:
  • Sections 1.1-5
  • none of Section 2 -- we'll do Section 5 instead
  • Sections 3.1-8 -- actually we blew off much of 3.3
  • Sections 4.1, 4.2, 4.4 -- we didn't explicitly have homework questions from 4.2, 4.4, but we did them a lot in class. See review questions below
  • Sections 5.1-4
  • Boyce & DiPrima (updated):
  • Sections 3.2, (less of 3.3), Nov. 13
  • Sections 4.1 Nov. 13
  • Sections 7.5-7.8 Nov 18,20
  • The stuff below will not be on the test
  • Sections 10.1, .2, .5 (not really 10.3, .4) Nov 25,Dec 2

    • Update: the above sections of [BDP] are the ones standardly taught in this course -- I'm following the generic math department recommendation for what to teach. What I'm hearing, though, is that the material from Ch1&2 was only covered at the end and extremely briefly (if at all) in 16b, 1b. So I am not going to assume that material; in particular, if we have any inhomogeneous differential equations to solve (and we might not) I will tell you their explicit form and solution.

    Here's a calendar for the class. It includes what we've done so far, and where we plan to go next.

    Homework:

  • Due 9/2: 1.1 #13,14,15, 1.2 #7,8,9,12,26,27,28,29. Warning:This homework is too easy, and not necessarily representative of the homework to come.
  • Due 9/9:
  • 1.3 #9-12, 15, 16
  • 1.4 #3-6, 16-20
    •
  • Due 9/16:
  • 1.5 #2, 3, 40 ("unit" lower triangular means 1s on the diagonal),
  • If {v,w} is an orthonormal set of vectors, and x and y are real numbers, for which x,y is {xv,yw} again an orthonormal set?
  • What are all the vectors orthogonal to (1,0,-1)?
  • 3.1 #41-44, 47-50
    •
  • Due 9/23:
  • 3.1 #21,22
  • 3.2 #1-2, 16-17, 19, 36-37
  • Let T be an 4x4 matrix, upper triangular, with 0s on the diagonal. Show that T^4, that means T*T*T*T, is the zero matrix.
  • Let W be like the identity matrix, but with the 1s on the NE/SW diagonal instead of NW/SE.
  • If U is upper triangular, show that WUW is lower triangular.
  • If U is both upper and lower triangular, is U = WUW?
  • Show W is invertible, by finding its inverse.
  • Midterm review problems (some too easy to be indicative)
  • from the worksheets, Questions/Problems/Additional
  • section 2 Q#2,4,5 / A#1
  • section 3 Q#1,2 / P#1,3,5,6,7 / A#1,2
  • section 4 Q#1,2 / P#3,4,5 / A#1,2
  • section 6 Q#2,4 / P#1,5 / A#1
  • Let T be a 10x10 square matrix that differs from the identity only in the top row and rightmost column, and has 1s in the NW and SE corners. Describe T*T. If 10x10 is too scary do 4x4 first. Then find T inverse. Hint: it's the same shape as T.
  • (If you've played with complex numbers before) Say we used the same definition of dot product, x.y = sum_i x_i y_i, for vectors full of complex numbers not just real numbers. Find a size two vector that is orthogonal to itself (!). (This is why we won't eventually use the same definition for dot product.)
  • Say that M,N are two matrices of the same width, each in row-echelon form. Stack M on top of N, making a tall matrix.
  • When is this in row-echelon form?
  • To put it in row-echelon form, do we have to change M sometimes?
  • To put it in row-echelon form, do we have to use M to change N sometimes?
  • To put it in row-echelon form, do we have to use N to change N sometimes?
  • Let M,N be rectangular such that MN, NM both make sense, and call them S,T. Say S=identity. Is T=identity? If not, what can you say about T^2?
  • Let {v,w} be an orthonormal pair of vectors in R^3, 3-dimensional space. How many vectors are there such that {v,w,x} is orthonormal?
  • This week's homework: 3.4 #1,3,4,6,7,8. The quiz will be on Friday rather than Wednesday.
  • Due 10/14: 3.4 #25,27,29,30, 3.5 #10,11,12,27,29,32,33
  • Due 10/21: 3.6 #7,8,9,11,26,35,36, 3.7 #33,34,35,43
  • Due 10/28: 3.8 #3,4,5,13,14,18,19,20,31
  • Second midterm review problems (some too easy to be indicative)
  • from the worksheets, Questions/Problems/Additional
  • section 8 Q#2,5 / P#2,3,4,5 / A#1,2
  • section 9 Q#3,4 / P#1,3,4,5 / A#1,2
  • section 10 Q#3 / P#1,2,4 / A#1,2,3
  • section 11 Q#1,2,4 / P#4,5,6,7
  • section 12 etc. are on stuff since the last homework, hence not on the test. for example section 12 P#1,3,6 / A#1,2,3 should not have been suggested for review! OOPS
  • Due 11/11: 4.1 #10,11,16, 5.1 #9,10,25,26,32,34,35
  • Due 11/18: [BDP] 3.2 #4,13,17, no actually forget this #17 -- it uses stuff that too many people haven't seen. 4.1 #11,14,17
  • Due 11/25: [BDP] 7.5 #1,2,7,11,15,29, 7.6 #1,2,9,13,28
  • Due 12/4: [Hill] 5.2 #16,17,23, 5.3 #8,9, [BDP] 7.8 #5,7,15,17
  • Final review problems
  • from the worksheets, Questions/Problems/Additional
  • section 12 Q#1,3,4 / P#2,6 / A#1,4
  • section 13 Q#1 / P#3,5
  • section 14 Q#1 / P#1,3 / A#1,3,4,5
  • While the above two sections didn't make it to homework, we did inner products on at least five occasions in class, so YES one might have to e.g. know the definition of orthogonal on the final. If you can do the review questions, you'll have all you need to know from 4.2,4.4.
  • section 15 Q#2,3 / P#1,3,4,5,6 / A#2,3 (may be hard!)
  • section 16 Q#1,2 / P#1,2,3,4,5,6 (cute question) / A#1
  • section 17 Q#1,2 / P#1,2,3,4 / A#1,3,4
  • section 19 Q#4,5 / P#1,2,3
  • section 20 Q#1 / P#1,4
  • section 21 Q#1
  • section 22 Q#1 / P#2,3 / A#1,2,3
  • section 23 Q#2 / P#1,2,5 / A#1
    • Errata from the final class:
  • I misstated what the solutions look like to a linear homogeneous system of ODEs when the eigenvalues repeat; where I had {1, t, t^2, t^3, ...} it should have been {1, t, t^2/2!, t^3/3!, ...}.
  • Worse, I claimed that the Wronskian test is if and only if! Check out e.g. section 19 problem #3.
    • Oops!