Exercise 1

For each of the following geometric operations in the plane, find a $2\times 2$ matrix that defines the matrix transformation performing the operation.

1. Rotates vectors by $180^\circ$.

2. Reflects vectors across the vertical axis.

3. Reflects vectors across the line $y=-x$.

4. Rotates vectors counterclockwise by $60^\circ$.

5. First rotates vectors counterclockwise by $60^\circ$ and then reflects in the line $y=x$.