Problem 1:
3.2 #46
Problem 2:
2.8 #38
Problem 3:
5.1 #21 (all parts)
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. 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 real eigenvalue, find
a basis for the corresponding eigenspace.
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.
Problem 5:
5.6 # 5 and 6
I'll be posting responses to select ASULearn messages I receive from the
class in the forum on ASULearn -
so ask/look on ASULearn for hints and suggestions.
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));
> 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)