### 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.)