Let T be the rotation about the origin of angle pi/2 followed by thereflection in the x-axis. Find T[1 0].

The transformation T consists of two steps: a rotation about the origin by an angle of pi/2, followed by a reflection in the x-axis.

Step 1: Rotation about the origin by an angle of pi/2 The rotation matrix for an angle of pi/2 is given by:

| cos(pi/2)  -sin(pi/2) |   =   | 0  -1 |
| sin(pi/2)   cos(pi/2) |       | 1   0 |

So, if we apply this rotation to the vector [1 0], we get:

| 0  -1 |   *   | 1 |   =   | 0 |
| 1   0 |       | 0 |       | 1 |

So, after the rotation, the vector [1 0] becomes [0 1].

Step 2: Reflection in the x-axis The reflection in the x-axis can be represented by the matrix:

| 1   0 |
| 0  -1 |

If we apply this reflection to the vector [0 1] (the result of the rotation), we get:

| 1   0 |   *   | 0 |   =   | 0 |
| 0  -1 |       | 1 |       |-1 |

So, after the reflection, the vector [0 1] becomes [0 -1].

Therefore, the result of the transformation T on the vector [1 0] is [0 -1].