Exercise 5

In this example and this example, we wrote matrix transformations in terms of the components of $T(\xvec)$. This exercise makes use of that form.

1. Let's return to the example in this activity concerning the company that operates two bakeries. We used a matrix transformation with input $\xvec$, which recorded the amount of time the two bakeries operated, and output $T(\xvec)$, the number of cakes, doughnuts, and eclairs produced. The associated matrix is $A=\begin{bmatrix} 10 & 20 \\ 50 & 30 \\ 30 & 30 \\ \end{bmatrix}$.

   1. If $\xvec=\twovec{x_1}{x_2}$, write the output $T(\xvec)$ as a three-dimensional vector in terms of $x_1$ and $x_2$.

   2. If Bakery 1 operates for $x_1$ hours and Bakery 2 for $x_2$ hours, how many cakes are produced?

   3. Explain how you may have discovered this expression by considering the rates at which the two locations make cakes.

2. Suppose that a bicycle sharing program has two locations $P$ and $Q$. Bicycles are rented from some location in the morning and returned to a location in the evening. Suppose that

   • 60% of bicycles that begin at $P$ in the morning are returned to $P$ in the evening while the other 40% are returned to $Q$.

   • 30% of bicycles that begin at $Q$ are returned to $Q$ and the other 70% are returned to $P$.

   1. If $x_1$ is the number of bicycles at location $P$ and $x_2$ the number at $Q$ in the morning, write an expression for the number of bicycles at $P$ in the evening.

   2. Write an expression for the number of bicycles at $Q$ in the evening.

   3. Write an expression for $T\left(\twovec{x_1}{x_2}\right)$, the vector that describs the distribution of bicycles in the evening.

   4. Use this expression to identify the matrix $A$ associated to the matrix transformation $T$.