Math 2240

Dr. Sarah's Math 2240 Calendar - Spring 2011

Office hours this week
ASULearn To access ASULearn (messages, solutions, and grades).
Syllabus and Grading Policies
Maple Tips and Access
Class highlights If you miss a class, then check here and make up the work before the next class.

Jump down to tomorrow's homework which is located above the red lines

Date	WORK DUE at the beginning of class or lab unless otherwise noted!
30 Apr - Sat	Final Project Poster Sessions from 12-2:30. Chess: Rena Juang Conway's Game of Life: Dawn Woodard Cramer's Rule: Amber Cox Cryptography: Chandra Houck, Kim Kiser and Peter Taylor Digital Imaging and Graphics: Miles Harris and Ryan Harris Eigenfaces and Facial Recognition: Thomas Beardsley, Spencer Gilbert and Brian Hancock Eight Queens Problem: Taylor Askew and Courtney Howlett Financial Banking: Michael Bulmer Game Theory: Lauren Alston History: Victoria Christian Least Squares Solutions: Dino Karic Linear Programming: Lucas Bozeman Mathematics Biology: Meredith Branham Neural Networks: Ryan Bennett and Mike Hancock NFL Quarterback Ranking: Tiffany Bert Quantum Mechanics: David Monroe Singular Value Decomposition: Robert Powers
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23 Apr - Tues	Work on the final project and write down any questions you have.
21 Apr - Thur	Test 3 study guide. Meet in 205. Test 2 revisions for a possible +3 for correct revisions. Turn in your original test with the revisions.
19 Apr - Tues	Meet in 205. Study for test 3 and write down any questions you have. Also work on test 2 revisions and continue working on the final project.
14 Apr - Thur	Problem Set 6 See Problem Set Guidelines and Sample Problem Set Write-Ups Note: You may work with two other people and turn in one per group of three Maple Commands and Hints for PS 6 Problem 1: 7.1 #26 by hand and on Maple via the Eigenvectors(A); command also compare your answers and resolve any apparent conflicts or differences. Problem 2: 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: 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? (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.) Part B: For each real eigenvalue find a basis for the corresponding eigenspaces. Part C: Use only a geometric explanation to explain why most rotation matrices have no real eigenvectors (ie keeping it on the same line through the origin). Problem 3: 7.2 7 Problem 3: 7.2 14 Problem 3: 7.2 20 Problem 6: Foxes and Rabbits (Predator-prey model) Suppose a system of foxes and rabbits is given as: F_k+1 = .55F_k + .45R_k R_k+1 = -.125F_k + 1.05R_k Part A: 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. Part B: Use the decomposition to explore 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, and then fill in the blanks: If ___ equals 0 then we die off along the line____ [corresponding to the eigenvector____], and otherwise we [choose one: die off or grow or hit and then stayed fixed] along the line____ [corresponding to the the eigenvector____]. Part C: Determine a value to replace 1.05 in the original system that leads to constant levels of the fox and rabbit populations (ie an eigenvalue of 1), so that eventually neither population is changing. What is the ratio of the sizes of the populations in this case?
7 Apr - Thur	Start thinking about a topic for the Final Project and if you are ready to select one, then message me on ASULearn for approval [first come-first-served]. Begin working on Problem 1 and Problem 6 on the problem set.
5 Apr - Tues	Read over the Final Project criteria
31 Mar - Thur	Test 2 on Chapters 1-3 and 4. study guide Test 1 revisions are due for a possible +4 (if they are correct - I am happy to look over them in office hours prior to the test)
29 Mar - Tues	Read over Problem Set 4 and 5 solutions on ASULearn. Examine the study guide and if you have any questions then write them down (to turn in).
24 Mar - Thur	Problem Set 5 See Problem Set Guidelines and Sample Problem Set Write-Ups Note: You may work with two other people and turn in one per group. Hints and Commands for Problem Set 5 Problem 1: 4.4 15 Problem 2: 4.4 54 Problem 3: 4.5 38 Problem 4: 4.5 48 Problem 5: Cement Mixing Continued (ALL IN MAPLE) This problem is worth more than the others Problem 6: 4.6 26 Problem 7: 4.6 30
22 Mar - Tues	Work on all but the 4.6 problems on problem set 5. Work on test 1 revisions.
17 Mar - Thur	Practice Problems (to turn in) 4.4 11, 53 4.5 22 Reminder that this Friday, March 18 is the last day to drop a class- I will update your total absences on ASULearn by Thursday.
15 Mar - Tues	Problem Set 4 Problem Set 4 See Problem Set Guidelines and Sample Problem Set Write-Ups Hints and Commands for Problem Set 4 Problems 1: If possible, write v=Matrix(([1],[-2],[-2]]) as a linear combination of u₁=Matrix(([2],[1],[3]]), u₂=Matrix(([1],[3],[5]]), and u₃=Matrix(([3],[4],[8]]) In addition, is v in the same geometric space that the 3 other vectors form under linear combinations? Explain what geometric space this is and why or why not. Problem 2: 4.1 44 Problem 3: *Cement Mixing (ALL IN MAPLE - including text comments)* This problem is worth more than the others. Problems 4-8: If it is a vector space/subspace, justify why by quoting the book or class, but if not, write out a complete proof via counterexamples, ie, what violating it means, and where each step follows logically from the previous step, like in class: Problem 4:* Even Numbers Prove that the set of even numbers is not a vector space (addition and scalar multiplication as usual). Problem 5: True or False: The line x-2y=0 is a vector space (addition and scalar multiplication as usual). Problem 6: Solutions to the plane 5x-3y+6z=11, ie {(x,y,z) in R³ so that 5x-3y+6z=11} Prove that this is not a subspace of R³ using axiom 1 (addition as usual). Problem 7: 4.3 (14 part E - Be sure to leave n as general as in class - do not define it as 2x2 matrix). Prove that this is not a subspace of the set of nxn matrices. Problem 8: 4.3 16. If it is not a subspace then prove this using axiom 6.
3 Mar - Thur	4.2 21 [Show that axiom 1 is violated, ie find two determinant 0 matrices that sum to a matrix with determinant non-zero] Continue working on Problem Set 4
1 Mar - Tues	Practice Problems (to turn in) 4.1 For 35, Part a) First solve this algebraically. Part b) Then, as in Chapter 4, plot the columns of the coefficient matrix using commands like: with(plots): col1:=spacecurve({[2t,3t,5t,t=0..1]}): display(col1,col2, col3) Do the columns of the coefficient matrix lie in the same plane? If not then they will generate all of 3-space under linear combinations, so anything will be a linear combination of them. If they do, then you can add the Matrix([[10],[1],[4]]) vector into the spacecurve and display command to see if it also lies in that plane. Part C): As in chapter 1, plot the three rows of the augmented matrix for the system using commands like row1:=implicitplot3d({2x+y-2*z-10},x=-20..20,y=-20..20,z=-20..20, color=yellow): display(row1,row2,row3) Are the rows lines or planes and how do they intersect [no common intersection, a single point, an entire line, or an entire plane] Then do #43 Begin working on the first 3 problems of problem set 4
24 Feb - Thur	Practice Problems (to turn in) 4.1 2, 4, 7 and 52.
22 Feb - Tues	Test 1 on Chapters 1, 2 and 3 in room 205 study guide
17 Feb - Thur	Read over Problem Set 1, 2, and 3 Solutions on ASULearn and examine the study guide under Tuesday's test date. Write down (to turn in) any questions you have.
15 Feb - Tues	Problem Set 3 Problem Set 3 See Problem Set Guidelines, Sample Problem Set Write-Ups Note: You may work with at most two other people and turn in one per group. Maple Commands and Hints for PS 3 I also encourage you to ask me questions about anything you don't understand in office hours or message me on ASULearn. Your group's explanations must distinguish your work as your own. Problem 1: 2.5 24 Problem 2: Healthy/Sick Workers (all on Maple including text comments) This problem is worth more than the others. Problem 3:* 3.1 47 part a Problem 4: 3.2 32 part c Problem 5: 3.3 (28 by-hand and on Maple) Problem 6: 3.3 (34 if a unique solution to Sx=b exists, find it by using the method x=MatrixInverse(S).b in Maple). Problem 7: 3.3 (50 parts a & c)
10 Feb - Thur	Proctice Problem: Do 3.3 number 31 by hand using a determinant calculation Continue working on Problem Set 3
8 Feb - Tues	Practice Problems in Chapter 2.5 Part A Set up the stochastic matrix N for the system in 2.5 number 10: The first column of N represents A->A, A->B, and A->Neither [.75, .20, .05 is the first column; .75, .15, .10 is the first row]. Part B Using regularity, we can see that the system will stabilize since the columns add to 1, and the entries are all positive. Find the steady-state vector by setting up and solving (N-I)x=0 for x. Recall that if you add a row of 1s at the bottom, this will solve for the value you want [the entries add to 100%]. Reduce in Maple, but be sure to put in fractions instead of decimals. Begin working on the Healthy/Sick worker problem on the next problem set.
3 Feb - Thur	Review Problem Set 2 solutions on ASULearn
1 Feb - Tues	Problem Set 2 - See Problem Set Guidelines and Sample Problem Set Write-Ups Note: You may work with at most two other people and turn in one per group. Maple Commands and Hints for PS 2 I also encourage you to ask me questions about anything you don't understand in office hours or message me on ASULearn. Your group's explanations must distinguish your work as your own. Problem 1: 2.1 30 Problem 2: 2.2 34 parts a, b & c Problem 3: 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. Problem 4: 2.3 12 Problem 5: 2.3 14 by hand and on Maple Problem 6: 2.3 28 part a - write the matrix system as Ax=b and apply the inverse method of solution in Maple Problem 7: 2.3 40 part d
27 Feb - Thur	Practice Problems in 2.1 and 2.2: (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 methods. Do not worry about explaining your work. 2.1 (by-hand: 9, 32) 2.2 (by-hand: 17, 18), (35 parts b and c)
25 Jan - Tues	Read through 1.2 practice solutions on ASULearn. Problem Set 1 - See Problem Set Guidelines and Sample Problem Set Write-Ups. Note: You may work with at most two other people and turn in one printed copy per group. 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 message me on ASULearn. Your group's explanations must distinguish your work as your own. Problem 1: 1.1 60 part c Problem 2: 1.1 74 Problem 3: 1.2 30 by hand and also on Maple Problem 4: 1.2 32 Problem 5: 1.2 44 parts a) through d) - in b) and d) find all the values of k and justify Problem 6: 1.3 24 parts a and b Problem 7: 1.3 26
20 Jan - Thur	Compare your 1.1 practice problems with solutions on ASULearn. A similar style of explanation is necessary for problem set 1 but not for practice problems. Do these practice problems by-hand since you need to get efficient at the by-hand method: 1.2 25, 27, and (43 - find all the values of k and justify why these are all of them). 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. Continue working on problem set 1 under the due date of Jan 25th.
18 Jan - Tues	Practice Problems to turn in - answers to odd problems are in the back of the book and a student solution manual is in Math Lab 1.1 55, (59 parts b and c), and 73. Use by-hand Gaussian Elimination on 55 and 73. For the 59 true/false questions, if a part is false, provide a specific counterexample, if it is true, quote a phrase from the text. Begin working on problem set 1 - the problems are listed under the due date of Jan 25th.
13 Jan - Thur	Read through the online syllabus carefully. Search google for Dr. Sarah, click on my page, and click on the MAT 2240 link and then the Syllabus link. Prepare to share something you read there and write down any questions you have - the university considers this a binding contract between us. Obtain the textbook and i-clicker available for rental at the bookstore. Bring the i-clicker to all classes. Practice Problems to turn in - answers to odd problems are in the back of the book and a student solution manual is in Math Lab 1.1 7, 15, 19 Don't worry about getting the correct answer - instead concentrate on the ideas and the methods. This will count as participation and will not receive a specific grade, although I will mark whether you reasonably attemped the problems.