### 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:**
Chapter 3 supplementary exercises **#18 on p. 187**. [Note: by the "results of
Exercise 16," the book means the line about 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:** When you solve for Nul A, include the definition of Null A
in your explanation/annotated reasoning.

**Part B:** One method (Method 2 from class) for Col A:
Circle the pivots and
provide the pivot columns of 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.
Examine any inconsistent parts
(like [0 0 0 0 combination of bs]) and set the equal column of these 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) (plug in each and see if it equals 0)

**Part E:** What is the geometry of Col A? (use part B to answer
what the geometry is and choose one from
[point, line, plane, hyperplane, entire space] and explain why in your
annotations)

**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 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 (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 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)