If \(A\) is the matrix \(A=\left[\begin{array}{rr} \vvec_1 \amp \vvec_2
\end{array}\right]\text{,}\) what is \(T\left(\twovec{0}{1}\right)\) in terms of the vectors \(\vvec_1\) and \(\vvec_2\text{?}\)
Suppose \(T: \IR^3 \rightarrow \IR^2\) is a linear map, and you know \(T\left(\left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right] \right)
=
\left[\begin{array}{c} 2 \\ 1 \end{array}\right]\) and \(T\left(\left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right] \right)
=
\left[\begin{array}{c} -3 \\ 2 \end{array}\right]
\text{.}\) What piece of information would help you compute \(T\left(\left[\begin{array}{c}0\\4\\-1\end{array}\right]\right)\text{?}\)
Write the matrix \([T(\evec_1) \,\cdots\, T(\evec_n)]\) for \(T\text{.}\) The matrix \(A\) is called the standard matrix for the transformation \(T\text{.}\)
Subsection2.5.1Applications of matrix transformations
Activity2.5.9.
Suppose that we work for a company that produces baked goods, including cakes, donuts, and eclairs. Our company operates two plants, Plant 1 and Plant 2. In one hour of operation,
Plant 1 produces 10 cakes, 50 donuts, and 30 eclairs.
If plant 1 operates for \(x_1\) hours and Plant 2 for \(x_2\) hours, how many cakes \(C\) does the company produce? How many donuts \(D\text{?}\) How many eclairs \(E\text{?}\)
We combine the number of hours the two plants operate into a vector \(\xvec=\twovec{x_1}{x_2}\text{.}\) Likewise, we use a vector \(\threevec{C}{D}{E}\) to denote the number of cakes \(C\text{,}\) donuts \(D\text{,}\) and eclairs \(E\) our company produces.
Now define a matrix transformation \(T(\xvec) =
\threevec{C}{D}{E}\) where \(\threevec{C}{D}{E}\) represents the number of baked goods produced when the plants are operated for times \(\xvec=\twovec{x_1}{x_2}\text{.}\) If \(T(\xvec)
= A\xvec\text{,}\) what are the dimensions of the matrix \(A\text{?}\)
Suppose the marketing department says we need to produce 1500 cakes, 4700 donuts, and 3300 eclairs. Is it possible to meet this order? If so, how long should the two plants operate?
We can consider the matrix transformation \(P(\xvec) = \twovec{F}{S}\) that tells us how many units of flour and sugar are required when we operate the plants for \(x_1\) and \(x_2\) hours. Find the matrix that defines the transformation \(P\text{.}\)
Suppose we run a company that has two warehouses, which we will call \(P\) and \(Q\text{,}\) and a fleet of 1000 delivery trucks. Every day, a delivery truck goes out from one of the warehouses and returns every evening to one of the warehouses. Every evening,
70% of the trucks that leave \(P\) return to \(P\text{.}\) The other 30% return to \(Q\text{.}\)
We will use the vector \(\xvec=\twovec{P}{Q}\) to represent the number of trucks at location \(P\) and \(Q\) in the morning. We consider the matrix transformation \(T(\xvec) = \twovec{P'}{Q'}\) that describes the number of trucks at location \(P\) and \(Q\) in the evening.
Suppose that all 1000 trucks begin the day at location \(P\) and none at \(Q\text{.}\) How many trucks are at each location at the end of the day? Therefore, what is the vector \(T\left(\ctwovec{1000}{0}\right)\text{?}\)
In the same way, suppose that all 1000 trucks begin the day at location \(Q\) and none at \(P\text{.}\) How many trucks are at each location at the end of the day? What is the result \(T\left(\ctwovec{0}{1000}\right)\text{?}\)
Suppose that there are 100 trucks at \(P\) and 900 at \(Q\) at the beginning of the day. How many are there at the two locations at the end of the day?
Suppose that there are 550 trucks at \(P\) and 450 at \(Q\) at the end of the day. How many trucks were there at the two locations at the beginning of the day?
Suppose that \(S\) is the matrix transformation that transforms the distribution of trucks \(\xvec\) one morning into the distribution of trucks two mornings later. What is the matrix that defines the transformation \(S\text{?}\)
Suppose that \(R\) is the matrix transformation that transforms the distribution of trucks \(\xvec\) one morning into the distribution of trucks one week later. What is the matrix that defines the transformation \(R\text{?}\)
