### 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:  Chapter 3 supplementary exercises #19 on p. 187.
A1:=Matrix([[1,1,1],[1,2,2],[1,2,3]]); Determinant(A1); will compute the determinant in Maple.
For the last part of the question, work by-hand. Keep n general so that the last row of your matrix is [0 1 2 3 ... n]. Use only replacement row operations (what will that do to the determinant?) and annotate/show what happens to the nxn matrix in each step of the reduction. Note that the book has partial hints on this since it is an odd problem.

Problem 2:  2.8 #26 with 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? (use part B to answer what the geometry is)

Problem 3:  5.6# 17 with modified directions:
Part A Part a. from the book. Be sure that your matrix A acts on the vector that has x as juveniles and y as adults (hint: juveniles = 0* prior juveniles + 1.6*female adults...).
Part B Part b. from the book. Don't forget to use fractions instead of decimals if you are using Eigenvectors(A); and/or evalf(Eigenvectors(A)); in Maple (your other option is by-hand work) and note that the book should have added "for most starting populations" somewhere in the text of b. Be sure to justify the growth and give the growth rate and the ratio. Annotate your reasoning.
Part C [Ignore the book] Roughly sketch a by-hand a trajectory plot with x as juveniles and y as adults, with starting population in the 1st quadrant 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 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? (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: For each theta that gives a 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 (but change the angle), 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)