### Problem Set 4

See the Guidelines and Maple Tips. 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:  Chapter 3 supplementary exercises #18 on p. 187. [Note: by the "results of Exercise 16," the book means the line just above 16a reading "Confirm that det A=(a-b)..." so plug a, b and n in there for each matrix. In addition, Determinant(A); will compute the determinant in Maple, which you are directed to do by the book]

Problem 2:  2.8 #25. Additional Instructions:
Part A: First solve for Nul A by parametrizing and show work. Include the definition of Null A in your explanation/annotated reasoning. You can use ReducedRowEchelonForm in Maple.
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,b4]]) and apply GaussianElimination(Augmented); in Maple. Are there any inconsistent parts (like [0 ... 0 combination of bs]) to set equal to 0?
Part D: Show that each basis column from your answer in part b) satisfies any equations that you obtained in part c)
Part E: What is the geometry of Col A? Why? Use Part B to answer what the geometry is and specify why---is it a point, line, plane, hyperplane?

Problem 3:  5.6 #5 with the following directions:
Part A: By-hand or in Maple compute the eigenvalues and eigenvectors. If you are in Maple, don't forget to use fractions in A instead of decimals if you are using Eigenvectors(A);. Also, notice that while p=.325 this means that (-.325) is in the matrix.
Part B: Write out the eigenvector decomposition for the system.
Part C: Explain why both populations grow for most starting populations (the book should have added this caveat after the word 'grow'.)
Part D: Answer the rest of the book's question: the growth rate and the ratio (note that x is owls and y is squirrels).
Part E: Roughly sketch by-hand a trajectory plot with x as owls and y as squirrels, with starting populations in the 1st quadrant that are not on either eigenvector, and that includes both eigenvectors in the sketch.

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, and you'll want this in your annotate.)
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)
> 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)