Exercise 13

This problem is a continuation of the previous problem.

  1. Let us define vectors

    \begin{equation*} \vvec_1 = \twovec{5}{2}, \vvec_2 = \twovec{-1}{1}\text{.} \end{equation*}

    Show that

    \begin{equation*} A\vvec_1 = \vvec_1, A\vvec_2 = -0.3\vvec_2\text{.} \end{equation*}
  2. Suppose that \(\xvec_1 = c_1 \vvec_1 + c_2 \vvec_2\) where \(c_2\) and \(c_2\) are scalars. Use the Linearity Principle expressed in Proposition 2.2.3 to explain why

    \begin{equation*} \xvec_{2} = A\xvec_1 = c_1\vvec_1 -0.3c_2\vvec_2\text{.} \end{equation*}
  3. Continuing in this way, explain why

    \begin{equation*} \begin{aligned} \xvec_{3} = A\xvec_2 \amp {}={} c_1\vvec_1 +(-0.3)^2c_2\vvec_2 \\ \xvec_{4} = A\xvec_3 \amp {}={} c_1\vvec_1 +(-0.3)^3c_2\vvec_2 \\ \xvec_{5} = A\xvec_4 \amp {}={} c_1\vvec_1 +(-0.3)^4c_2\vvec_2 \\ \end{aligned}\text{.} \end{equation*}
  4. Suppose that there are initially 500 bicycles at location \(B\) and 500 at location \(c\text{.}\) Write the vector \(\xvec_1\) and find the scalars \(c_1\) and \(c_2\) such that \(\xvec_1=c_1\vvec_1 + c_2\vvec_2\text{.}\)

  5. Use the previous part of this problem to determine \(\xvec_2\text{,}\) \(\xvec_3\) and \(\xvec_4\text{.}\)

  6. After a very long time, how are all the bicycles distributed?