Comparison of Linear Algebra in the High School and University Setting: by Candace and Melissa
Computer Programing with Matrices and Images: by Josh
Computer Programming with Vectors: by Bethany
Google's PageRank Algorithm: by Kaj
Eigenvectors with Deer Populations and Hunters: by Alana
Electronic Circuits: Isaac
Fractals and Linear Algebra: Ondrej
Java Program that Takes the Determinant of Matrices: Coty and Ryan
Linear Algebra and Contra Dancing: by Katie Mebane
Linear Algebra and Game Theory: Catherine
Linear Algebra and Graphic Design: Darrell
Linear Algebra and High School Math Education: by Caitlin and Katie
Linear Algebra in Hydrogeology: by Anna
Linear Algebra involved in Nash Equilibriums: by Jeremy and Justin
Linear Algebra & Parimutuel Wagering Systems: by Stephen
Linear Algebra and Sports Ranking: by Alex
Linear Algebra and Temperature Distribution: Austin and Marc
Matrices and Game Theory: by Amy and Philip
Matrices and Fibonacci Numbers: by Becca
Special Unitary Groups: by Davidson
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 explain 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.
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?
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
4.4 11, 53
4.5 22
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.
4.2 21 [Show that axiom 1 is violated, ie find two determinant 0 matrices that sum to a matrix with determinant non-zero]
4.1 35, 43, 52
4.1 7 and 49.
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=S^(-1) b)
Problem 7: 3.3 (50 parts a & c)
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 and the co-factor expansion method.
3.3 31
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%].
I have added a Stochastic/Markov System Demo in 2.5 (from class on 9/16) file so that you can review related content to help you.
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
2.1 (by-hand: 9, 32)
2.2 (by-hand: 17, 18), (35 parts b and c)
Note: You may work with at most two other people and turn in one 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 parts 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
1.2 25, 27, and (43 parts a) through d) - in parts b) and d) 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.
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.