### Problem Set 4

See the Guidelines. I will post on ASULearn answers to select questions I receive via messaging there or in office hours. I am always happy to help!

Mathematics, you see, is not a spectator sport. [George Polya, How to Solve it]

Problem 1:  3.2 #46 Determinant(A); will compute the determinant in Maple. You will have 6 matrices.

Problem 2:  2.8 #24. Additional Instructions:
Part A: First solve for Nul A, and include the definition of Null A in your explanation/annotated reasoning.
Part B: Next solve for Col A as follows: Reduce A, circle the pivots and provide the pivot columns of A (not reduced A) as the basis for the Col A.
Part C: Find an equation that the vectors in Col A satisfy as follows: Set up and solve the augmented matrix for the system Ax=Vector([b1,b2,b3]]) and apply GaussianElimination(Augmented); in Maple. Are there any inconsistent parts (like [0 0 0 0 0 combination of bs]) to set equal to 0?
Part D: Show that each basis column in part b) satisfies the equation that you obtained in part c)
Part E: What is the geometry of Col A using part B? Choose one from [point, line, plane, volume, hyperplane, entire space, other] and briefly explain why in your annotations.

Problem 3:  5.6# 6. Additional Instructions:
Part A: In Maple compute the eigenvalues and eigenvectors using fractions in A instead of decimals using Eigenvectors(A); for p=1/2.
Part B: Write out the eigenvector decomposition for the system.
Part C: Explain why the populations die off.
Part D: For most starting positions, what is the yearly die off rate in the long term. Explain where your number came from.
Part E: For most starting positions, what is the eventual ratio the system tends to? Explain where your ratio came from.
Part F: Sketch (by-hand) a trajectory diagram for the system, by graphing the two eigenvectors, picking a starting point in the first quadrant different from the eigenvectors, and sketching what happens over time, like in the examples in the glossary on ASULearn (for trajectory).
Part G: Find a value of p for which the populations of both owls and squirrels tend toward constant levels and explain how you obtained p
Part H: What are the relative population sizes in this case? Explain where your numbers came from.

Problem 4:  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 (Eigenvalues, not Eigenvectors here) in Maple or solve for the eigenvalues by-hand. Notice that there are real eigenvalues for certain values of theta only. What are these values of theta and what eigenvalues do they produce? Show work/reasoning. (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--you'll want this as part of your annotations)
Part B: For each real eigenvalue, find a basis for the corresponding eigenspace (Pi is the correct way to express pi in Maple - you can use comamnds like Eigenvectors(Matrix([[cos(Pi/2),-sin(Pi/2)],[sin(Pi/2),cos(Pi/2)]])); in Maple, or by-hand otherwise.
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). Address the definition of eigenvalues/eigenvectors in your response as well as how the rotation angle connects to the definition in this case.

A Review of Various Maple Commands:
> with(LinearAlgebra): with(plots):
> A:=Matrix([[-1,2,1,-1],[2,4,-7,-8],[4,7,-3,3]]);
> ReducedRowEchelonForm(A);
> GaussianElimination(A);
(only for augmented matrices with unknown variables like k or a, b, c in the augmented matrix)
> Transpose(A);
> ConditionNumber(A);
(only for square matrices)
> Determinant(A);
> Eigenvalues(A);
> Eigenvectors(A);
> evalf(Eigenvectors(A));
decimal approximation
> Vector([1,2,3]);
> B:=MatrixInverse(A);
> A.B;
> A+B;
> B-A;
> 3*A;
> A^3;
> evalf(M);
> spacecurve({[4*t,7*t,3*t,t=0..1],[-1*t,2*t,6*t,t=0..1]},color=red, thickness=2);
plot vectors as line segments in R3 (columns of matrices) to show whether the the columns are in the same plane, etc.
> implicitplot({2*x+4*y-2,5*x-3*y-1}, x=-1..1, y=-1..1);
> implicitplot3d({x+2*y+3*z-3,2*x-y-4*z-1,x+y+z-2},x=-4..4,y=-4..4,z=-4..4);
plot equations of planes in R^3 (rows of augmented matrices) to look at the geometry of the intersection of the rows (ie 3 planes intersect in a point, a line, a plane, or no common points)