### 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 *n*x*n* 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 R**^{2}
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 R^{3}
(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)