Exercise 13

This problem is a continuation of the previous problem.

1. Let us define vectors $$\vvec_1 = \twovec{5}{2}, ~~~ \vvec_2 = \twovec{-1}{1}$$. Show that $$A\vvec_1 = \vvec_1, ~~~A\vvec_2 = 0.3\vvec_2$$.

2. Suppose that $\xvec_1 = c_1 \vvec_1 + c_2 \vvec_2$ where $c_1$ and $c_2$ are scalars. Use the Linearity Principle expressed in Linearity of matrix multiplication to explain why $$\xvec_{2} = A\xvec_1 = c_1\vvec_1 + 0.3c_2\vvec_2$$.

3. Continuing in this way, explain why $$\begin{aligned} \xvec_{3} = A\xvec_2 & {}={} c_1\vvec_1 +0.3^2c_2\vvec_2 \\ \xvec_{4} = A\xvec_3 & {}={} c_1\vvec_1 +0.3^3c_2\vvec_2 \\ \xvec_{5} = A\xvec_4 & {}={} c_1\vvec_1 +0.3^4c_2\vvec_2 \\ \end{aligned}$$.

4. Suppose that there are initially 500 bicycles at location $B$ and 500 at location $C$. Write the vector $\xvec_1$ and find the scalars $c_1$ and $c_2$ such that $\xvec_1=c_1\vvec_1 + c_2\vvec_2$.

5. Use the previous part of this problem to determine $\xvec_2$, $\xvec_3$ and $\xvec_4$.

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