Question

To every linear transformation T from R2 to R2, there is an associated 2×2 matrix. Match the following linear transformations with their associated matrix.

1. Clockwise rotation by π/2 radians
2. Reflection about the x-axis
3. Counterclockwise rotation by π/2 radians
4. The projection onto the x-axis given by T(x,y) = (x,0)
5. Reflection about the y-axis e
6. Reflection about the line y = x
A. (-1 0, 0 1)
B. (0 1, 1 0)
C. (1 0, 0 0)
D. (0 1, -1 0)
E. (1 0, 0 -1)
F. (0 -1, 1 0)
G. None of the above

Answer 1

1. Clockwise rotation by π/2 radians

: D

2. Reflection about the x-axis

: E

3. Counterclockwise rotation by π/2 radians

: F

4. The projection onto the x-axis given by T(x,y) = (x,0)

: C

5. Reflection about the y-axis : A

6. Reflection about the line y = x : B

Explanation:

Given a linear transformation
`T:\mathbb{R}^2\to\mathbb{R}^2`, one matrix representation of T is obtained by stacking the vectors T(1,0) and T(0,1) in columns.

a) A counter clockwise rotation of
`\pi/2` radians sends (1,0) to (0,-1) and it sends (0,1) to (1,0), so the matrix representation is

`\left[\begin{matrix} 0& 1 \\ -1 & 0 \end{matrix}\right]` which corresponds to matrix D.

From now on, I will provide the values of T(1,0) and T(0,1)

b) Reflection about the x-axis T(1,0) = (1,0) and T(0,1) = (0,-1), which corresponds to matrix E.

c) Counterclockwise rotation by π/2 radians T(1,0) = (0,1), T(0,1) = (-1,0). Matrix F

d) The projection onto the x-axis given by T(x,y) = (x,0). T(1,0) = (1,0) T(0,1) = (0,0). Matrix C

e) Reflection about the y-axis T(1,0) = (-1,0) T(0,1) = (0,1). Matrix A

f) Reflection about the line y = x. This transformation corresponds to interchanging the values of x and y. That is, send (x,y) to (y,x). So, in this case

T(1,0) = (0,1) T(0,1) = (1,0). Matrix B