### Problem Set 4

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

Problem 1:  3.1 #44 Determinant(A); will compute the determinant in Maple, and RandomMatrix(4); will give you a 4x4 matrix with integer entries.

Problem 2:  2.8 #37. Additional Instructions:
Part A: When you solve for Nul A, include the definition of Null A in your explanation/annotated reasoning.
Part B: One method for Col A: Reduce A, circle the pivots and provide the pivot columns of A (not reduced A) as the basis for the Col A.
Part C: Another method (Method 1 from class) for Col A: 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 0 0 0 combination of bs]) to set equal to 0?
Part D: What is the geometry of Col A using part B? Choose one from [point, line, plane, hyperplane, entire space] and explain why in your annotations.

Problem 3:  5.6 #4 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);.
Part B: Write out the eigenvector decomposition for the system.
Part C: Do the populations grow, die off, stabilize, or exibit some other behavior in the longrun for most starting conditions? Explain.
Part D: For most starting positions, what is the yearly rate in the long term (growth rate, die off rate, or stability rate) and the eventual ratio the system tends to? Explain where your numbers came from.
Part E: 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 demos on ASULearn or in this urban-rural populations example of the eigenvector decomposition, the critical reasoning, and the trajectory diagram.
Part F: What do you think might happen to the system if some aspect of the model (such as birth rates or the predation rate) were to change slightly?

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)
> Transpose(A);
> ConditionNumber(A);
(only for square matrices)
> Determinant(A);
> Eigenvalues(A);
> Eigenvectors(A);
> evalf(Eigenvectors(A));
> 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);
> display(a,b,c);
> 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)