What is the dimension of the vectors \(\xvec\) that are inputs for \(T\text{?}\)

(b)

What is the dimension of the vectors \(T(\xvec)=A\xvec\) that are outputs?

(c)

Describe the vectors \(\xvec\) for which \(T(\xvec) = \zerovec\text{.}\)

Activity2.5.0.3.

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 \(\vvec_2\text{?}\)

Activity2.5.0.4.

Suppose that \(A\) is a \(3\times 2\) matrix and that \(T(\xvec)=A\xvec\text{.}\) If

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 \(T\left(\left[\begin{array}{c}0\\4\\-1\end{array}\right]\right)\text{?}\)

The value of \(T\left(\left[\begin{array}{c} 0\\-4\\0\end{array}\right]\right)\text{.}\)

The value of \(T\left(\left[\begin{array}{c} 0\\1\\0\end{array}\right]\right)\text{.}\)

The value of \(T\left(\left[\begin{array}{c} 1\\1\\1\end{array}\right]\right)\text{.}\)

Any of the above.

Activity2.5.0.7.

Let \(T: \IR^4 \rightarrow \IR^3\) be the linear transformation given by

Write the matrix \([T(\evec_1) \,\cdots\, T(\evec_n)]\) for \(T\text{.}\) The matrix \(A\) is called the standard matrix for the transformation \(T\text{.}\)

Activity2.5.0.8.

Let \(T: \IR^3 \rightarrow \IR^2\) be the linear transformation given by

\begin{equation*}
T\left(\left[\begin{array}{c} x\\ y \\ z \end{array}\right] \right) = \left[\begin{array}{c} x+3z \\ 2x-y-4z \end{array}\right]
\end{equation*}

(a)

Compute \(T(\evec_1)\text{,}\)\(T(\evec_2)\text{,}\) and \(T(\evec_3)\text{.}\)

(b)

Find the standard matrix for \(T\text{,}\) so that \(T(\xvec) =
A\xvec\text{.}\)

Subsection2.5.1Applications of matrix transformations

Activity2.5.1.1.

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.

Plant 2 produces 20 cakes, 30 donuts, and 30 eclairs.

(a)

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 \(E\text{?}\)

(b)

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 \(A\text{?}\)

(c)

Find the vector \(T\left(\twovec{1}{0}\right)\) and the vector \(T\left(\twovec{0}{1}\right)\) and use your results to write the matrix \(A\text{.}\)

(d)

If we operate Plant 1 for 40 hours and Plant 2 for 50 hours, how many baked goods have we produced?

(e)

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?

Activity2.5.1.2.

Continue with the same baked good producing plants from the previous activity. Consider the needed ingredients:

Each cake requires 4 units of flour and and 2 units of sugar.

Each donut requires 1 unit of flour and 1 unit of sugar.

Each eclair requires 1 units of flour and 2 units of sugar.

(a)

Suppose we make \(C\) cakes, \(D\) donuts, and \(E\) eclairs. How many units of flour \(F\) are required? How many units of sugar \(S\text{?}\)

(b)

Write a matrix \(B\) that defines the matrix transformation \(R\left(\threevec{C}{D}{E}\right) =
\twovec{F}{S}\text{.}\)

(c)

If Plant 1 operates for 30 hours and Plant 2 operates for 20 hours, how many units of flour and sugar are required?

(d)

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 \(P\text{.}\)

Activity2.5.1.3.

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 \(Q\text{.}\)

50% of the trucks that leave \(Q\) return to \(Q\) and 50% return to \(P\text{.}\)

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.

(a)

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 \(T\left(\ctwovec{1000}{0}\right)\text{?}\)

Using this result, what is \(T\left(\twovec{1}{0}\right)\text{?}\)

(b)

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 \(T\left(\ctwovec{0}{1000}\right)\text{?}\)

(c)

Find the matrix \(A\) such that \(T(\xvec) =
A\xvec\text{.}\)

(d)

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?

(e)

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?

Activity2.5.1.4.

Continue with the same trucking company from the previous activity.

Suppose that all of the trucks are at location \(Q\) on Monday morning.

(a)

How many trucks are at each location Monday evening?

(b)

How many trucks are at each location Tuesday evening?

(c)

How many trucks are at each location Wednesday evening?

Activity2.5.1.5.

Continue with the same trucking company from the previous two activities.

(a)

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 \(S\text{?}\)

(b)

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 \(R\text{?}\)

(c)

What happens to the distribution of trucks after a very long time?

All the trucks end up at location \(P\text{.}\)

All the trucks end up at location \(Q\text{.}\)

625 trucks end up at location \(P\text{,}\) while 375 end up at location \(Q\text{.}\)

375 trucks end up at location \(P\text{,}\) while 625 end up at location \(Q\text{.}\)