Preview Activity 2.6.1

Suppose that we wish to describe the geometric operation that reflects 2-dimensional vectors in the horiztonal axis. For instance, FigureĀ 2.6.1 illustrates how a vector \(\xvec\) is reflected into the vector \(T(\xvec)\text{.}\)

Figure 2.6.1 A vector \(\xvec\) and its reflection \(T(\xvec)\) in the horizontal axis.
  1. If \(\xvec = \twovec{2}{4}\text{,}\) what is the vector \(T(\xvec)\text{?}\) Sketch the vectors \(\xvec\) and \(T(\xvec)\text{.}\)

  2. More generally, if \(\xvec=\twovec{x}{y}\text{,}\) what is \(T(\xvec)\text{?}\)

  3. Find the vectors \(T\left(\twovec{1}{0}\right)\) and \(T\left(\twovec{0}{1}\right)\text{.}\)

  4. Use your results to write the matrix \(A\) so that \(T(\xvec) = A\xvec\text{.}\) Then verify that \(T\left(\twovec{x}{y}\right)\) agrees with what you found in part b.

  5. Describe the transformation that results from composing \(T\) with itself; that is, what is the transformation \(T\circ T\text{?}\) Explain how matrix multiplication can be used to justify your response.