##### Activity2.6.3

In this activity, we seek to describe various matrix transformations by finding the matrix that gives the desired transformation. All of the transformations that we study here have the form $$T:\real^2\to\real^2\text{.}$$

1. Find the matrix of the transformation that has no effect on vectors; that is, $$T(\xvec) = \xvec\text{.}$$ We call this matrix the identity and denote it by $$I\text{.}$$

2. Find the matrix of the transformation that reflects vectors in $$\real^2$$ over the line $$y=x\text{.}$$

3. What is the result of composing the reflection you found in the previous part with itself; that is, what is the effect of reflecting in the line $$y=x$$ and then reflecting in this line again. Provide a geometric explanation for your result as well as an algebraic one obtained by multiplying matrices.

4. Find the matrix that rotates vectors counterclockwise in the plane by $$90^\circ\text{.}$$

5. Compare the result of rotating $$90^\circ$$ and reflecting in the line $$y=x$$ to the result of first reflecting in $$y=x$$ and then rotating $$90^\circ\text{.}$$

6. Find the matrix that results from composing a $$90^\circ$$ rotation with itself. Explain the geometric meaning of this operation.

7. Find the matrix that results from composing a $$90^\circ$$ rotation with itself four times; that is, if $$T$$ is the matrix transformation that rotates vectors by $$90^\circ\text{,}$$ find the matrix for $$T\circ T\circ T \circ T\text{.}$$ Explain why your result makes sense geometrically.

8. Explain why the matrix that rotates vectors counterclockwise by an angle $$\theta$$ is

\begin{equation*} \left[\begin{array}{rr} \cos\theta \amp -\sin\theta \\ \sin\theta \amp \cos\theta \\ \end{array}\right] \text{.} \end{equation*}
in-context