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 by \(90^\circ\) and then 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