## Dr. Sarah's Math 2240 Web Page - Summer 2006

• Office Hours: Thur 6/29 2:40 - , Fri 6/30 12-12:40
• Campus Pipeline To access WebCT (bulletin board, solutions, and grades). Direct WebCT access if Pipeline is down
• Sakai access if WebCT is down
 Date WORK DUE at the beginning of class or lab unless otherwise noted! June 30 - Fri Test 3 revisions due. Final Project Poster Sessions   Peer Review   Self Evaluation   Session Info Aaron Grizzly Bear Population Derrick Moore-Penrose and Neural Networks Evan Structural Geology and Brachiopods Heather NFL Ratings Jason LU Decomposition Joe Astronomical Digital Image Processing with Linear Algebra Josh History of Linear Algebra Kevin Fluid Mechanics Mandy R. Fractals Mandy S. Quantum Mechanics and Schrodingers Equation Monique NFL Ratings Tommy Programming Row Operations in Java Veronica Linear Algebra Spans the Sciences __________ ________________________________________________________________________ __________ ________________________________________________________________________ June 29 - Thur Final project abstract due by 12 noon as an attachment that I can read (text, Word, rtf, or Maple) June 28 - Wed Test 2 revisions due for a possible +5 added onto your grade for complete and correct revisions. Turn in your original test along with the revisions. Test 3 study guide June 27 - Tues Final project proposal (a short description of what you plan to do) and preliminary list of references due. Your topic needs to be pre-approved as there is a limit to the number of people per topic. Study for test 3 and write down any questions you have. June 26 - Mon Problem Set 6 - See Problem Set Guidelines and Sample Problem Set Write-Ups *7.1 #14 by hand and on Maple via the Eigenvectors(A); command -- also compare your answers and resolve any apparent conflicts or differences within Maple text comments. *Rotation matrices in R2 Recall that the general rotation matrix which rotates vectors in the counterclockwise direction by angle theta is given by M:=Matrix([[cos(theta),-sin(theta)],[sin(theta),cos(theta)]]); Part A:Use only a geometric explanation to explain why most rotation matrices have no eigenvalues or eigenvectors. Part B: Apply the Eigenvalues(M); command. Notice that there are real eigenvalues for certain values of theta only. What are these values of theta and what eigenvalues do they produce? Find a basis for the corresponding eigenspaces. (Recall that I = the square root of negative one does not exist as a real number and that cos(theta) is less than or equal to 1 always.) *7.2 7, 18, and 24 *Foxes and Rabbits (Predator-prey model) Suppose a system of foxes and rabbits is given as:  Write out the Eigenvector decomposition of the iterate x_k, where the foxes F_k are the first component of this state vector, and the rabbits R_k the second. Use the decomposition to explain what will happen to the vector x_k in the longterm, and what kind of vector(s) it will travel along to achieve that longterm behavior. Extra Credit: Determine a value of the [2, 2] entry that leads to constant levels of the fox and rabbit populations, so that eventually neither population is changing. What is the ratio of the sizes of the populations in this case? June 23 - Fri Test 1 revisions for a possible +5 added onto your grade for complete and correct revisions. Turn in your original test along with the revisions. Test 2 on Chapters 1-3 and 4 study guide June 22 - Thur Review for test 2 and write down any questions you have. Read over the final project links under the June 30 due date. Work on Test 1 revisions and begin working on Problem Set 6. June 21 - Wed Problem Set 5 - See Problem Set Guidelines, Sample Problem Set Write-Ups, and Hints for Problem Set 5 4.4   16 4.5   24, 48 Cement Mixing Continued (**ALL IN MAPLE**) This problem is worth more than the others 4.6   24, 27 Jun 20 - Tues Practice Problems (to turn in) 4.4   11, 53 4.5   22 4.6   22, 31 Work on Problem Set 5 June 19 - Mon Problem Set 4 See Problem Set Guidelines and Sample Problem Set Write-Ups, and Hints and Commands for Problem Set 4 4.1 36 and 44 Cement Mixing (*ALL IN MAPLE*) *This problem is worth more than the others. For all of the following vector space and subspace problems: If it is a vector space or subspace, then just state that it is, but if it is not, then write out the complete proof that one axiom is violated as in class: 4.2   22 Natural Numbers   Prove that the natural numbers is not a vector space using axiom 6. True or False:   The line x+y=0 is a vector space. Solutions to 2x-3y+4z=5, ie {(x,y,z) in R^3 so that 2x-3y+4z=5}   Prove that this is not a subspace of R3 using axiom 1. 4.3   (14 part D be sure to leave n as general - do not define it as 2) Extra Credit   Prove that the subset of R^5 consisting of all the solutions of the nonhomogenous equation Ax=b, where A is a given 4x5 matrix and b is a given non-zero vector in R^4 is not a subspace Jun 16 - Fri Test 1 on Chapters 1, 2 and 3 study guide Jun 15 - Thur Practice Problems (to turn in) 4.1   7, 35, 43, 49, 52, 4.2   21 June 14 - Wed Problem Set 3 See Problem Set Guidelines, Sample Problem Set Write-Ups, and Hints and Commands for Problem Set 3 2.5   24 Healthy/Sick Workers (all on Maple) *This problem is worth more than the others. 3.1   47 a 3.2   32 c 3.3   (28 by-hand and on Maple), (34 if a unique solution to Sx=b exists, find it by using the method x=S^(-1) b), and (50 a and c) June 13 - Tues Practice Problems (to turn in). Do not worry about getting the same answer as the back of the book (although it would be nice!) but do concentrate instead on making sure you understand the determinant methods. 2.5   10 set-up the stochastic matrix and calculate month 1 only. 3.1   33 by-hand using the co-factor expansion method. 3.2   25 by-hand using some combination of row operations and the co-factor exapansion method. 3.3   31 Work on PS 3 Jun 12 - Mon Problem Set 2 - See Problem Set Guidelines, Sample Problem Set Write-Ups, and Maple Commands and Hints for PS 2. 2.1   30 2.2   (34 parts a, b and c) Show that the following statements about matrices are false by producing counterexamples and showing work:       Statement a) A2=0 implies that A = 0       Statement b) A2=I implies that A=I or A=-I       Statement c) A2 has entries that are all greater than or equal to 0. 2.3   12, (14 by hand and on Maple), (28 part a - write out the matrix system as Ax=b and then apply the inverse method of solution), and (40 part d) Jun 9 - Fri Practice Problems (to turn in) 2.1   (7 use matrix algebra and equality to obtain a system of 4 equations in the 3 unknowns and then solve), (by-hand: 9, 11, 15, 32) 2.2   17, 18, (35 use matrix algebra to combine the elements, set it equal to the other side, use matrix equality to obtain equations, and solve.) Work on Problem Set 2 Jun 8 - Thur Read through Sample Problem Set Write-Ups Problem Set 1 - See Problem Set Guidelines, Sample Problem Set Write-Ups, and Problem Set 1 Maple Commands and Hints. I also encourage you to ask me questions about anything you don't understand in office hours or on the WebCT bulletin board. Your explanations must distinguish your work as your own. 1.1   (24 on Maple), 60 b and c, 74, 1.2   (30 by hand and also on Maple), 32, (44 find all values of k and justify), (59 find one example that works) 1.3   24, 26 June 7 - Wed Compare your 1.1 practice problems with solutions on WebCT. A similar style of explanation is necessary for problem set 1 but not for practice problems. Do these by-hand since you need to get efficient at the by-hand method Show work and be prepared to turn this in and/or present but no need to write in complete sentences. 1.2   15, 25, 27, (43 find all the values of k and justify why these are all of them), and 49. Do not worry about getting the same answer as the back of the book (although it would be nice!) but do concentrate instead on making sure you understand the method of Gaussian Elimination. Read through Problem Set Guidelines and Problem Set 1 Maple Commands and Hints Continue working on problem set 1. June 6 - Tues Read through the online syllabus carefully and write down any questions you have - the university considers this a binding contract between us. Do the following by-hand since you need practice: The answers to odd problems are in the back of the book and there is a student solution manual in mathlab (M-Th, 2-5) - it is your job to make sure you understand the process/work and could present it. Show work and be prepared to turn this in, but no need to write in complete sentences. 1.1   7, 15, 19, 55, (59 parts b and c - if it is false, provide a counter-example), and 73. Begin working on problem set 1.