Date  WORK DUE at the beginning of class or lab unless otherwise noted! 
8 Aug  Fri 
Cramer's Rule: Lamonte Cryptography: David Eigenvectors and Social Security: Joseph Image Edge Detection and Linear Algebra: Michael Linear Algebra in Biology: Elizabeth and Jessica Linear Algebra in the Gaming Industry: Brandon Matrices of the Banking World: Jack Matrices and Music: Casey and Rebecca Matrix Multiplication using Parallel Processors/MultiCore Computers: John Network Analysis: Matt NFL Ratings: Amy and Douglas 
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7 Aug  Thur 

6 Aug  Wed 

5 Aug  Tues 

4 Aug  Mon 
Note: You may work with two other people and turn in one per group of three Problem 1: 7.1 #14 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? Also 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.) Part B: Use only a geometric explanation to explain why most rotation matrices have no eigenvalues or eigenvectors. Problem 3: 7.2 7 Problem 4: 7.2 18 Problem 5: 7.2 24 Problem 6: Foxes and Rabbits (Predatorprey model) Suppose a system of foxes and rabbits is given as: 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 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. 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? 
31 July  Thur 

29 July  Tues 
Note: You may work with two other people and turn in one per group. Hints and Commands for PS 5 Problem 1: 4.4 16 Problems 2: 4.5 24 Problems 3: 4.5 48 Problem 4: Cement Mixing Continued (**ALL IN MAPLE**) This problem is worth more than the others Problems 5: 4.6 24 Problems 6: 4.6 27 
25 July  Fri 
4.4 11, 53 4.5 22 
23 July  Wed 
Note: You may work with two other people and turn in one per group. Hints and Commands for Problem Set 4 Problems 1: 4.1 36 Problem 2: 4.1 44 Problem 3: Cement Mixing (*ALL IN MAPLE*) *This problem is worth more than the others. Problem 4: 4.2 22 Problem 5: Natural Numbers Prove that the natural numbers is not a vector space using axiom 6. Problem 6: True or False: The line x+y=0 is a vector space. Problem 7: Solutions to the plane 2x3y+4z=5, ie {(x,y,z) in R^3 so that 2x3y+4z=5} Prove that this is not a subspace of R^{3} using axiom 1. Problem 8: 4.3 (14 part D Be sure to leave n as general as in class  do not define it as 2x2 matrix). Prove that this is not a subspace. 
22 July  Tues 
4.2 21 
21 July  Mon 

18 July  Fri 
4.1 7, 35, 43, 49, 52 
17 July  Thur 
Note: You may work with at most two other people and turn in one per group. Maple Commands and Hints for PS 3 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=S^(1) b) Problem 7: 3.3 (50 parts a & c) 
16 July  Wed 
2.5 number 10. The first column of N represented A>A, A>B, and A>Neither [.75, .20, .05 is the first column; .75, .15, .10 is the first row]. 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 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%]. 3.1 33 byhand using the cofactor expansion method. Expand along the first column to take advantage of the 0s, and then the 1st column of the next 4x4 matrix, and then the 3rd row of the 3x3 matrix. 3.2 25 byhand using some combination of row operations and the cofactor exapansion method. 3.3 31 
15 July  Tues 
Maple Commands and Hints for PS 2 Note: You may work with at most two other people and turn in one per group but each person must complete and turn in Problem 5 themselves (in their own words). 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  look at the matrix system as Ax=b and then apply the inverse method of solution Problem 7: 2.3 40 part d 
14 July  Mon 
2.1 (byhand: 9, 32) 2.2 (byhand: 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 using the ReducedRowEchelonForm command on Maple. No need to print out your Maple work  just summarize it on your homework.) 
11 July  Fri 
Note: You may work with at most two other people and turn in one per group but each person must complete and turn in Problem 3 themselves (in their own words). 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 
10 July  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. 
9 July  Wed 
1.1 7, 15, 19, (59 parts b and c  if a part is false, provide a specific counterexample, if it is true, quote a phrase from the text), and 73. 