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. B 1. The projection onto the x-axis given by T(x,y)=(x,0) A 2. Counter-clockwise rotation by π/2 radians C 3. Clockwise rotation by π/2 radians A 4. Reflection about the y-axis B 5. Reflection about the x-axis F 6. Reflection about the line y=x A. (−1001) B. (1000) C. (100−1) D. (0−110) E. (01−10) F. (0110) G. None of the above

Answer 1

Answer:

1. B

2. D

3. E

4. A

5. C

6. F  

Step-by-step explanation:  

1. The projection onto the x-axis is given by T(x, y) = (x, o)  =(1 0 0 0) B  

2. Counter-clockwise rotation by π/2 radians C

= (0 - 1 1 0) D  

3. Clockwise rotation by π/2 radians

= (0 1 - 1 0) E  

4. Reflection about the y-axis

= (-1 0 0 1) A  

5. Reflection about the x-axis  = (1 0 0 - 1) C  

6. Reflection about the line y=x

(0 1 1 0) F  

For every line in a plane, there is a linear transformation that reflects the vector about that line. The easiest way to answer a question like this is to figure out where the standard basic vector is, e1 and e2. Write the answers at the column of the matrix. Letting As be the matrix corresponding to the linear transformation s. It is easier to see that e1 gets carried to e2 and e2 gets carried to - e1

As= (0 - 1 1 0)