##### Exercise13

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 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?

in-context