Matrix transformations

In this activity, we will look at some examples of matrix transformations.

1. To begin, suppose that $A$ is the matrix with associated matrix transformation $T(\xvec) = A\xvec$.

   1. What is $T\left(\twovec{1}{-2}\right)$?

   2. What is $T\left(\twovec{1}{0}\right)$?

   3. What is $T\left(\twovec{0}{1}\right)$?

   4. Is there a vector $\xvec$ such that $T(\xvec) = \twovec{3}{0}$?

   5. Write $T\left(\twovec xy\right)$ as a two-dimensional vector.

2. Suppose that $T(\xvec) = A\xvec$ where .

   1. What is the dimension of the vectors $\xvec$ that are inputs for $T$?

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

   3. If we des…

Suppose we run a company that has two warehouses, which we will call $P$ and $Q$, and a fleet of 1000 delivery trucks. Every morning, a delivery truck goes out from one of the warehouses and returns in the evening to one of the warehouses. It is observed that

• 70% of the trucks that leave $P$ return to $P$. The other 30% return to $Q$.

• 50% of the trucks that leave $Q$ return to $Q$ and 50% return to $P$.

The distribu…

Suppose that $T$ is the matrix transformation defined by the matrix $A$ and $S$ is the matrix transformation defined by $B$ where .

1. If $T:\real^n\to\real^m$, what are the values of $m$ and $n$? What values of $m$ and $n$ are appropriate for the transformation $S$?

2. Evaluate $T\left(\threevec{1}{-3}{2}\right)$.

3. Evaluate $S\left(\threevec{-2}{2}{1}\right)$.

4. Evaluate $S\circ T\left(\threevec{1}{-3}{2}\right)$.

5. Find the matrix $C$ that defines the matrix transformation $S\circ T$.

This problem concerns the identification of matrix transformations, about which more will be said in the next section.

1. Check that the following function $T:\real^3\to\real^2$ is a matrix transformation by finding a matrix $A$ such that $T(\xvec) = A\xvec$. .

2. Explain why is not a matrix transformation.

Suppose that the matrix defines the matrix transformation $T:\real^3\to\real^3$.

1. Describe the vectors $\xvec$ that satisfy $T(\xvec) = \zerovec$.

2. Describe the vectors $\xvec$ that satisfy $T(\xvec) = \threevec{-8}{9}{2}$.

3. Describe the vectors $\xvec$ that satisfy $T(\xvec) = \threevec{-8}{2}{-4}$.

Determine whether the following statements are true or false and provide a justification for your response.

1. A matrix transformation $T:\real^4\to\real^5$ is defined by $T(\xvec) = A\xvec$ where $A$ is a $4\times5$ matrix.

2. If $T:\real^3\to\real^2$ is a matrix transformation, then there are infinitely many vectors $\xvec$ such that $T(\xvec) = \zerovec$.

3. If $T:\real^2\to\real^3$ is a matrix transformation, then it is possible that every equation $T(\xvec) = \bvec$ has a solution for every vector $\bvec$.

4. If $T:\real^n\to\real^m$ is a matrix transformation, t…

Suppose that a company has three plants, called Plants 1, 2, and 3, that produce milk $M$ and yogurt $Y$. For every hour of operation,

• Plant $1$ produces 20 units of milk and 15 units of yogurt.

• Plant $2$ produces 30 units of milk and 5 units of yogurt.

• Plant $3$ produces 0 units of milk and 40 units of yogurt.

1. Suppose that $x_1$, $x_2$, and $x_3$ record the amounts of time that the three plants are operated and that $M$ and…

Suppose that $T:\real^2\to\real^2$ is a matrix transformation and that .

1. Find the vector $T\left(\twovec{1}{0}\right)$.

2. Find the matrix $A$ that defines $T$.

3. Find the vector $T\left(\twovec{4}{-5}\right)$.

Suppose that two species $P$ and $Q$ interact with one another and that we measure their populations every month. We record their populations in a state vector $\xvec = \twovec{p}{q}$, where $p$ and $q$ are the populations of $P$ and $Q$, respectively. We observe that there is a matrix such that the matrix transformation $T(\xvec)=A\xvec$ is the transition function describing how the state vector evolves from month to month. We also observe that…

Students in a school are sometimes absent due to an illness. Suppose that

• 95% of the students who attend school will attend school the next day.

• 50% of the students who are absent one day will be absent the next day.

We will record the number of present students $p$ and the number of absent students $a$ in a state vector $\xvec=\twovec{p}{a}$ and note that that state vector evolves from one day to the next according to the…