### 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 #43
Determinant(A); will compute the determinant in Maple,
and
A:=RandomMatrix(5); will give you a 5x5 matrix with
random integer entries.

**Problem 2:**
2.8 #12. **Additional Instructions**:

**Part A:** First solve for Nul A, include the definition of Null A in your
explanation/annotated reasoning, and then give *p* as directed.

**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. Then give *q* as directed.

**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,b5]]) 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, volume, hyperplane, entire space] and explain why in your annotations.

**Problem 3:**
5.6 #12 with the following directions:

**Part A:** In Maple compute the eigenvalues and
eigenvectors using **fractions** in A instead of decimals 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 why.

**Part D:**
For most starting positions, what is the yearly rate in the long term (growth rate,
die off rate, or stability rate). Explain where your number came from.

**Part E:** For most starting positions, what is
the eventual ratio the system tends to? Explain where your ratio came from.

**Part F:** 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 glossary on ASULearn (for trajectory).

**Part G:**
In Maple input the original matrix using
**decimals** and execute Eigenvectors(A);.
Are the eigenvectors approximately on the same lines the corresponding ones with fractions were?
Check each eigenvector and show work/reasoning.

**Part H:** Are the eigenvectors exactly on the same lines? (Hint: look at the last
digits of the *x* and *y* values in the eigenvector corresponding to the larger eigenvalue).

**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)**
**

> 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 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);

> 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)