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[My office is 741 Evans | My office hours are Tuesday and Friday from 12:30-2, or by appointment.] 
It's Summer 2008; I'm teaching Math 54. We meet for this class every weekday afternoon from 2-4 in 87 Evans. This website is where you can find all you need to know about this course. I'll post homework assignments, quiz solutions, and other useful info here. Check this page often!
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Course Materials: Week 1: [M 6/23][Syllabus] [Tu 6/24] [W 6/25][quiz 1] [Th 6/26] [F 6/27][hw1] Week 2: [M 6/30] [Tu 7/1][hw2] [W 7/2][quiz 2] [Th 7/3][hw3] [F 7/4] no school Week 3: [M 7/7] [Tu 7/8] [W 7/9][hw4][quiz 3] [Th 7/10][ch 3&4] [F 7/11][m1][m1 sols] Week 4: [M 7/7] [Tu 7/8] [W 7/9][quiz 4] [Th 7/10] [F 7/11] Week 5: [M 7/21] [Tu 7/22] [W 7/23][quiz 5] [Th 7/24][worksheet] [F 7/25] Week 6: [M 7/28] [Tu 7/29][sample m2] [W 7/30][quiz 6] [Th 7/31] [F 8/1][m2] Week 7: [M 8/4] [Tu 8/5] [W 8/6] [Th 8/7] [F 8/8] Week 8: [M 8/11][sample final] [Tu 8/12] [W 8/13] |
[posted 22 August 2008:] Final grades Here's the final gradesheet. The two lowest HW's were dropped, and the lowest quiz was dropped. In the end, the flat grading scale of 90, 80, 70, 60 was the best option. But if you were close to the cutoff or got some bonus points (the column labeled "b" is the bonus question from the final) then I was able to bump some of you up. All in all I was very pleased with your grades! I think you guys learned a lot, and hopefully had even half as much fun as I did. Adios! Oh, and here's the solutions to the final. [posted 15 August 2008:] Grades, pre-final, kind of Here's a spreadsheet of your grades prior to the final exam. The column labeled "grade" is your option 1 average, and the colum labeled "exam" is the average of your two midterms. Your lowest quiz is dropped. Due to technical difficulties this gradesheet only has your first 4 homeworks so far, so none of those are dropped in this version. I'll update it to include all your homeworks as soon as possible! Sorry about that. [posted 14 August 2008:] Final prep II I posted summaries of the last 3 weeks of class under the course materials list. We're having our review Thursday from noon-2 in 5 Evans, the usual spot. Then in class time we'll do more review--so that's 4 hours of review time for you guys. For good theoretical practice from the old book, try the suggested supplementary exercises from before each midterm (these are posted on the course calendar), as well as the super long true/false questions from Ch1-4. For good theoretical review from the new book, check out the chapter summaries at the end of each chapter. These give a nice concise take on all the crazy techniques we learned. Here's a list of some new skills you may be tested on: First, for a single linear differential equation: (1) Solving a homogeneous 2nd order differential equation with constant coefficients. (2) Using undetermined coefficients to find a particular solution of a nonhomogeneous DE. (3) Using variation of parameters to find a particular solution of a nonhomogeneous DE. (4) Solving an initial value problem for any of these. (5) Finding intervals on which a solution must exist for a DE with nonconstant coefficients. (6) Properly wielding the Wronskian. Next, for a first-order differential system: (1) Converting higher-order DEs into a 1st order system in normal form. (2) Finding solutions to a diagonalizable homogeneous differential system using the e-value/e-vector formula. (3) Finding a fundamental matrix of a diagonalizable system using exponentiation. (4) Properly wielding the Wronskian. (5) Graphing trajectories of a 2x2 differential system. (6) Using variation of parameters to find a particular solution to a nonhomogeneous differential system. Bonus: Fourier series: (1) Computing the Fourier coefficients of a periodic function. The exam will be about 60% new stuff and 40% old stuff. At least some of the old stuff will come directly from old quizzes and HWs. [posted 11 August 2008:] Final prep I posted some sample questions that would be good for the final. There's probably more to come. [updated all the time:] Course Calendar [posted 31 July 2008:] Midterm 2 more prep (scroll down; there's lots of stuff here) First off, here are some T/F explanations from the supplementary exercises at the end of each chapter: Chapter 5: (a) True: reciprocals of e-values for A form the e-values for A^-1. (b) False: invertible does not imply diagonalizable. (c) True: a row of zeros means A is not invertible, which means 0 is an e-value. (d) False: e-values of A^2 are the squares of the e-values of A. (e) True: if Av=kv, then (A^2)v=A(Av)=A(kv)=k(Av)=(k^2)v. (f) True: if Av=kv, then (A^-1)Av=(A^-1)kv. Divide by k and voila. (g) False: they can be 0. (h) True. (i) False: any e-space of dimension 2 or higher will have a basis consisting of more than just 1 linearly independent e-vector. (j) True: they have the same characteristic equation (because they have the same det.). (k) False: suppose Bv=kv and A=SB(S^-1). If you try to get anywhere with Av=SB(S^-1)v, you won't be able to move the v inside the (S^-1). (There's probably a better explanation for this.) (l) False: only true if they belong to the same e-space. (m) False: the zero entries count too. (n) True: because they have the same determinant. (o) False: you can have repeated e-values with full-dimensional e-spaces. (p) True: most rotations, for example. (q) False: diagonalizable does not imply invertible. (r) True: eigenspaces intersect only at the origin. (s) False: this is another attempt at trying to say diagonalizable is the same as invertible. (t) True: we have a basis of eigenvectors, so we can write A=SD(S^-1). But notice that S=I, so A=D. (u) True: suppose A=PB(P^-1) and B=SD(S^-1). Then A=PSD(S^-1)(P^-1)=(PS)D(PS)^-1. (v) True: AB=(B^-1)BA(B^-1)^-1. (w) False: some of those e-vectors could correspond to e-value 0. (x) True: diagonalizable means we can form a basis of e-vectors. Chapter 6: (I'll use square brackets instead of pointy brackets for inner products because html doesn't like pointy bracket symbols.) (a) True: part of axiom 4 of inner products. (b) True: [-v,-v] = (-1)(-1)[v,v]. (c) True: definition. (d) False: if r is negative you'll pull out its absolute value instead. (e) False: one of them might be the zero vector. (f) True: furthermore x is orthogonal to Span{u,v}. (g) True: [u+v,u+v]=[u,u]+2[u,v]+[v,v]=[u,u]+[v,v] implies that [u,v]=0. (h) True: same as (g). (i) False: it's a scalar multiple of u. (j) True: we proved this in class. (And it should make some intuitive sense: think shadows.) (k) True: in particular it's the orthogonal complement. (It's not too hard to prove it's a subspace.) (l) False: they have the zero vector in common. (m) True: multiplying by scalars doesn't change that all the inner products are 0. (n) False: it's the other way around. If you do it this way, then you have a tall skinny matrix times a short fat matrix, and that will multiply to give you a singular matrix (remember this? I called it the plumbing problem), so it can't equal I. (o) False: the columns have to be unit vectors. (p) True: we didn't learn this one. (q) True: this is a fancy way of stating the pythagorean theorem. (Think shadows.) (r) False: the "solution" is actually x-hat, not A(x-hat). (s) False: (we didn't define these things as the "normal equations," so don't worry about this one). The normal equations are actually (A^T)Ax=(A^T)b. Chapter 7: (a) True: that's the big theorem! (b) False: orthogonal means inverse = transpose, not matrix = transpose. (c) True: we proved this in class: [Ax,Ax]=(Ax)^T(AX)=(x^T)(A^T)AX=(x^T)(A^-1)AX=(x^T)x=[x,x]. (e) False: the columns must be unit vectors as well. Here's a list of some theroems and formulas you should know by memory (in no particular order): (1) The Cauchy-Schwarz inequality. (2) The spectral decomposition formula. (3) There exists a unique least squares solution of Ax=b iff the columns of A are linearly independent. (4) Two subspaces are orthogonal to each other iff their basis vectors are orthogonal to each other. (5) The inner product axioms (particularly the 4th axiom). (6) The following are equivalent for an nxn matrix A: -- A is symmetric. -- A is orthogonally diagonalizable. -- There's an orthogonal basis for R^n consisting of e-vectors of A. -- The e-spaces of A are all of full dimension and are all orthogonal to one another. (7) The geometric interpretation of eigenvectors (things that get merely stretched or squished by a matrix) and eigenvalues (the factor by which this stretching or squishing occurs). (8) The distance formula. And here's a list of some stuff that will be provided for you on the exam: (1) A wordy definition of the triangle inequality. (2) A vague description of the spectral decomposition formula. (3) The solution to the least squares problem Ax=b lies in solving (A^T)Ax=(A^T)b. All right, email me with your questions. Don't forget about emergency office hours at 1pm. Good luck people. [posted 30 July 2008:] Midterm 2 prep There's going to be a review session Thursday from noon to 2 in 5 Evans. I've also posted a sample midterm for you to try, as well as a list of good theoretical problems from the supplementary exercises to keep you sharp. Please check these out--at least one of these supplementary exercise problems will show up on the midterm! As far as computational skills go, here's a list of some of the main things you should be able to do: (1) You should be able to find eigenvalues, eigenvectors, and eigenspaces of a matrix. (2) You should know how to diagonalize a matrix. (3) You should be able to compute inner products of different kinds and perform Gram-Schmidt with them. (4) You should be able to compute orthogonal projections and find distances between vectors and subspaces. (5) You should know the Cauchy-Schwarz and triangle inequalities, and the formula for the angle between two vectors. (6) You should be able to solve least-squares problems. (7) You should be able to multiply and invert block matrices. (8) You should know how to recognize a symmetric matrix, orthogonally diagonalize it, and write out its spectral decomposition. Whew! (Don't worry, we'll have time to review some of these things on Thursday--especially the more recent topics.) These are all skills that we've already mostly honed in the homework, and that is the best place to go to get practice if you're feeling rusty. Try some odd-numbered exercises from the book. But don't peek at the answers until you've properly toiled on your own! Good luck people. [updated 30 July 2008:] Box of Doom! If you do any of these things in a problem, the problem gets a zero! This box will grow as the semester progresses. Nominations are welcome.
[posted 24 July 2008:] Midterm 1 data Here's how the problems were graded for midterm 1, out of a total of 48 possible points: problem 1 - (a)5, (b)2 problem 2 - (a)2, (b)2, (c)2, (d)2 problem 3 - (a)5, (b)2 problem 4 - (a)5, (b)2, (c)2 problem 5 - (a)5, (b)2, (c)2 problem 6 - (a)4, (b)4 Here's the grade breakdown: 45-48 A+ 42-44 A 40-41 B+ 34-39 B 30-33 C+ 23-29 C 20-22 D 0-19 F The the 25th percentile was 28, the median was 33, the 75th percentile was 41, and the max was 46. [posted 5 July 2008:] The writings on the wall We sure had fun the other day doing groupwork. [1] [2] [3] [4] [posted 23 June 2008:] Homeworks and Quizzes Hello folks. We'll have homework due every Tuesday and Friday, except for exam Fridays. Homework assignments will be posted on the course calendar below (the first 2 are up already). Feel free to make friends and work on the homework together, just make sure you write up your solutions on your own. Homework will be checked for completeness (which includes neatness) and a couple of problems will be chosen at random to be graded for correctness. We'll have a short quiz every Wednesday (even on exam weeks). Quiz problems should resemble stuff you'll see on the homework, and may also include vocabulary questions. |
[Last modified: 21 August 2008]