## Dr. Sarah's Math 2240 Web Page - Summer 2009

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 Date WORK DUE at the beginning of class or lab unless otherwise noted! 7 Aug - Fri Final Project Session. peer evaluation, self evaluation Linear Combination Efficiency in Glass Blowing: Mackenzie Fractals and Linear Algebra: Frank Graph theory and its Applications to Linear Algebra: Matt History of Linear Algebra: Brice Linear Algebra and Genetics - Punnet Square: Clay and Tim Merlin's Magic Squares and Linear Algebra: Stephanie L. and Travis Strassen Algorithm: JL Sudoku and Linear Algebra: Stephanie D. Visualization of Linear Algebra in CG: Jack Test 3 revisions for a possible +5. Turn in your original test too. 6 Aug - Thur Prepare to present your Final Project Abstract orally in class. Be sure that your topic is pre-approved as a message on ASULearn. __________ ________________________________________________________________________ __________ ________________________________________________________________________ 5 Aug - Wed Test 3 study guide Test 2 revisions for possible +3. Turn in your original test. 4 Aug - Tues Study for test 3 via the study guide and write down any questions you have. Work on test 2 revisions. Begin thinking about a final project topic. 3 Aug - Mon 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 Hints and Commands for Problem Set 6 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 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: Find a basis for the corresponding eigenspaces.   Part C:   Use only a geometric explanation to explain why most rotation matrices have no eigenvalues or eigenvectors (ie scaling along the same line through the origin). Problem 3:  7.2   7 Problem 4:  7.2 18 Problem 5:  7.2 24 Problem 6:  Foxes and Rabbits (Predator-prey model) Suppose a system of foxes and rabbits is given as:  Part A: Write out the Eigenvector decomposition of the iterate xk, where the foxes Fk are the first component of this state vector, and the rabbits Rk the second.   Part B: Use the decomposition to explore what will happen to the vector xk 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? 30 July - Thur Test 2 on Chapters 1-3 and 4. study guide Test 1 Corrections for a possible +3 due. Turn in your original test with your corrections. 39 July - Wed Study for test 2 via the study guide and write down any questions you have. Work on test 1 revisions. 28 July - Tues 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 PS 5 Problem 1: 4.4   16 Problem 2: 4.5   24 Problem 3: 4.5   48 Problem 4: Cement Mixing Continued (**ALL IN MAPLE**) This problem is worth more than the others Problem 5: 4.6   24 Problem 6: 4.6   27 27 July - Mon Work on Problem Set 5 and Test 1 revisions. 23 July - Fri Practice Problems (to turn in) 4.4   11, 53 4.5   22 22 July - Thur Problem Set 4 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 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 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. 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 - Wed Practice Problem to turn in 4.2 number 21 [Show that axiom 1 is violated, ie find two determinant 0 matrices that sum to a matrix with determinant non-zero] 21 July - Tues Practice Problems (to turn in) 4.1 7, 35, 43, 49, and 52. On 35 and 43 also specify the geometry of the rows and columns. 20 July - Mon Meet in 205 Test 1 on Chapters 1, 2 and 3 study guide Bring your book to class. 16 July - Thur 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) 15 July - Wed Review the ASULearn demo for 2.5 on Stochastic/Markov systems. Practice Problems (to turn in): Read 2.5 number 10. Part A Set up the stochastic matrix N for the system. 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%]. 3.1   33 by-hand using the co-factor 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 by-hand using some combination of row operations of the form (ri'=constant*rj+ri) and the co-factor expansion method. Work on Problem Set 3. 14 July - Tues Meet in 205. 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 - look at the matrix system as Ax=b and then apply the inverse method of solution Problem 7: 2.3   40 part d 13 July - Mon 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) Work on Problem Set 2 10 July - Fri 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 bulletin board. Your explanations must distinguish your work as your own. 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 9 July - 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 by-hand since you need to get efficient at the by-hand method. No need to write in complete sentences. 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 and Problem Set 1 Maple Commands and Hints and continue working on problem set 1. 8 July - Wed Read through the online syllabus carefully. Search google for Dr. Sarah, click on her 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. Practice Problems to turn in - 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) 1.1   7, 15, 19, (59 parts b and c), and 73. 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 attemped the problems. For 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 under Friday's due date.