- Define T(v) = Av, where A is Matrix([[1,0],[0,-1]]). Then T(v)

a) reflects v about the y-axis.

b) reflects v about the x-axis.

c) rotates v clockwise pi/2 radians about the origin.

d) rotates v counterclockwise pi/2 radians about the origin.

e) none of the above

- Define T(v) = Av, where A is Matrix([[0,-1],[1,0]]). Then T(v)

a) reflects v about the y-axis.

b) reflects v about the x-axis.

c) rotates v clockwise pi/2 radians about the origin.

d) rotates v counterclockwise pi/2 radians about the origin.

e) none of the above

- Define T(v) = Av, where A is Matrix([[.5,.5],[.5,.5]]). Then the range of T, the set of outputs of the linear transformation, is

a) R^{2}and I have a good reason why

b) R^{2}but I am unsure of why

c) the y=x line but I am unsure of why

d) the y=x line and I have a good reason why

e) other

- To rotate a figure about a point (-2,3) we can
first translate to the origin, then rotate there, and
finally translate back again. This is expressed as:

a) (Trans_by (2,-3)).(Rot).(Trans_by (-2,3)). Object

b) (Trans_by (-2,3)).(Rot).(Trans_by (2,-3)). Object

c) both of the above

d) neither of the above

- Which of the following are true about linear transformations?

a) points go in as column vectors, and the first column of the transformation represents the output of the unit*x*-axis

b) we must use homogeneous coordinates (higher dimensional coordinate=1) if we want to use translations

c) they compose from right to left, like functions

d) more than one of a, b and c, but not all of them

e) all of a, b and c

- Let T: x ---> Ax be given as a linear transformation arising from a
square 2x2 matrix A. Assume that the set of all
outputs b (from Ax=b) is a line. What can we deduce?

a) The columns of A do not span R^{2}and I can think of an example

b) The columns of A do not span R^{2}but I can not think of an example

c) The columns of A do span R^{2}and I can think of an example

d) The columns of A do span R^{2}but I can not think of an example

- To keep a car on a curved race track, we can perform the
appropriate matrix operations in the following order:

a) (Rotate).(Translate_to_curve).car

b) (Translate_to_curve).(Rotate).car

c) Either a) or b) works

d) Neither a) nor b) works

- To turn a car so that it points in the direction of motion, we

a) define a unit vector in the direction of the velocity (tangent) of the curve by dividing by its norm so it won't change the size of the car

b) create an orthogonal vector to pair with it in a rotation matrix by creating a vector on a line with negative reciprocal slope (swap x and y and introduce a negative sign)

c) both of the above

- To rotate Yoda, who was given in row vectors as opposed to column vectors,
we made use of

a) a matrix that rotates about a line in 3-space

b) transpose of AB = transpose of B times the transpose of A

c) Both of the above

d) Neither a) nor b)

Solutions

1. b)

2. d)

3. d)

4. b)

5. e)

6. a)

7. b)

8. c)

9. c)