Section 7.1
Problems 8, 14, 18, 20, 24, 28
Section 7.2
Problems 12, 13, 35 - 38
Section 7.3
Problems
6, 8, 17, 18
Section 7.4
Problems
2-10 (even), 20, 22
Extra Problems
In class, I proved that if M is a 2 x 2 matrix and L is a line in R^2,
then when you apply M to each point of the line L, you end up with
another line M(L) in R^2. (Unless M is the zero matrix, in which case
it sends the line L to the point (0,0): M(L) = the origin.)
The reason I proved this is that otherwise you might think that the
matrix M sends a straight segment to a curved object --- it might send
the line to a figure 8 or to a parabola.
For the following three problems, start by doing them with a specific
line or plane and a specific matrix M. Figure out how it works for
your specific example and present your specific example in your
homework. Then explain how it had nothing to do with your specific
example by re-presenting the arguement for an arbitrary matrix, and
arbitrary line or plane.
Problem 1.
Prove that if M is a 2 x 2 matrix and L is a segment of a
line in R^2, then when you apply M to each point of the line L, you
end up with another segment of a line M(L) in R^2. (Unless M is the
zero matrix, in which case it sends the line-segment L to the point
(0,0,0))
Once you've proven Problem 1, you've proven what I was using in class
all this week --- if you know where two points A and B are mapped,
then you know where the segment connecting A and B is mapped.
Problem 2.
Prove that if M is a 3 x 3 matrix and L is a line in R^3,
then when you apply M to each point of the line L, you end up with
another line M(L) in R^3. (Unless M is the zero matrix, in which case
it sends the line L to the point (0,0,0): M(L) = the origin.)
(Hint: as before, the line L is of the form:
L = the set of points (x0,y0,z0) + t (a1,a2,a3),
where t varies from -infinity to infinity.)
Problem 3.
Prove that if M is a 3 x 3 matrix and P is a plane in R^3,
then when you apply M to each point of the plane P, you end up with
another plane M(P) in R^3. (Unless M is the zero matrix, in which case
it sends the plane P to the point (0,0,0): M(L) = the origin, or
unless M has a zero determinant and sends the plane P to a line
L.)
(Hint: you can write a plane in two ways:
first way:
P = the set of points that satisfy a x + b y + c z = d
for some constants a, b, c, and d
second way:
P = the set of points (x0,y0,z0) + t1 (a1,a2,a3) + t2 (b1,b2,b3)
where t1 and t2 vary from -infinity to infinity.)
For problems 1,2,3 only consider the case where
det(M) isn't 0. So you don't have to prove that it takes a plane
to a line in the 3x3 case.