Date  WORK DUE at the beginning of class or lab unless otherwise noted! 
June 29  Fri 

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June 28  Thur 

June 27  Wed 

June 26  Tues 

June 25  Mon 
Note: You may work with one other person and turn in one per group of two but each person must complete and turn in Problem 1 themselves (in their own words). 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 within Maple text comments. 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 35: 7.2 7, 18, and 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. 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 22  Fri 

June 21  Thur 

June 20  Wed 
Note: You may work with one other person and turn in one per group. Problem 1: 4.4 16 Problems 23: 4.5 24, 48 Problem 4: Cement Mixing Continued (**ALL IN MAPLE**) This problem is worth more than the others Problems 56: 4.6 24, 27 
June 19  Tues 
4.4 11, 53 4.5 22 
June 18  Mon 
Note: You may work with one other person and turn in one per group. Problems 12: 4.1 36 and 44 Problem 3: Cement Mixing (*ALL IN MAPLE*) *This problem is worth more than the others. Problems 47: 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 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  do not define it as 2) 
June 15  Fri  
June 14  Thur  4.1 7, 35, 43, 49, 52, 4.2 21 
June 13  Wed 
Note: You may work with one other person and turn in one per group of two Problem 1: 2.5 24 Problem 2: Healthy/Sick Workers (all on Maple) *This problem is worth more than the others. Problem 3: 3.1 47 a Problem 4: 3.2 32 c Problem 57: 3.3 (28 byhand 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 12  Tues 
2.5 10 setup the stochastic matrix and calculate month 1 only. 3.1 33 byhand using the cofactor expansion method. 3.2 25 byhand using some combination of row operations and the cofactor exapansion method. 3.3 31 
June 11  Mon 
Note: You may work with one other person and turn in one per group of two 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 and 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. Problems 47: 2.3 12, (14 by hand and on Maple), (28 part a  look at the matrix system as Ax=b and then apply the inverse method of solution), and (40 part d) 
June 8  Fri 
2.1 (7 use matrix algebra and equality to obtain a system of 4 equations in the 3 unknowns and then solve), (byhand: 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.) 
June 7  Thur 
Note: You may work with one other person and turn in one per group of two but each person must complete and turn in Problem 3 themselves (in their own words). Problems 12: 1.1 60 b and c, 74, Problems 35: 1.2 (30 by hand and also on Maple), 32, (44 find all values of k and justify), Problems 67: 1.3 24 a and b, 26 
June 6  Wed 
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. 
June 5  Tues 
1.1 7, 15, 19, (59 parts b and c  if it is false, provide a counterexample), and 73. 