Date  WORK DUE at the beginning of class or lab unless otherwise noted! 
30 Apr  Sat 
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 

21 Apr  Thur 

19 Apr  Tues 

14 Apr  Thur 
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 R^{2} 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 (Predatorprey 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 

5 Apr  Tues 

31 Mar  Thur 

29 Mar  Tues 

24 Mar  Thur 
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 

17 Mar  Thur 
4.4 11, 53 4.5 22 
15 Mar  Tues 
Hints and Commands for Problem Set 4 Problems 1: If possible, write v=Matrix(([1],[2],[2]]) as a linear combination of u_{1}=Matrix(([2],[1],[3]]), u_{2}=Matrix(([1],[3],[5]]), and u_{3}=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 48: 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 x2y=0 is a vector space (addition and scalar multiplication as usual). Problem 6: Solutions to the plane 5x3y+6z=11, ie {(x,y,z) in R^{3} so that 5x3y+6z=11} Prove that this is not a subspace of R^{3} 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 

1 Mar  Tues  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({[2*t,3*t,5*t,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 3space 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({2*x+y2*z10},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 
24 Feb  Thur 
4.1 2, 4, 7 and 52. 
22 Feb  Tues  
17 Feb  Thur 

15 Feb  Tues 
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 byhand 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 

8 Feb  Tues 
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 steadystate vector by setting up and solving (NI)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. 
3 Feb  Thur 

1 Feb  Tues 
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) A^{2}=0 implies that A = 0 Statement b) A^{2}=I implies that A=I or A=I Statement c) A^{2} 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 
2.1 (byhand: 9, 32) 2.2 (byhand: 17, 18), (35 parts b and c) 
25 Jan  Tues 
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 
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. 
18 Jan  Tues 
1.1 55, (59 parts b and c), and 73. Use byhand 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. 
13 Jan  Thur 
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. 