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
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?
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
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.
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)
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%].
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 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
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.
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.