Matrix multiplication and linear combinations

1. Find the matrix product .

2. Suppose that $A$ is the matrix . If $A\xvec$ is defined, what is the dimension of the vector $\xvec$ and what is the dimension of $A\xvec$?

3. A vector whose entries are all zero is denoted by $\zerovec$. If $A$ is a matrix, what is the product $A\zerovec$?

4. Suppose that $I = \left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array}\right]$ is the identity matrix and $\xvec=\threevec{x_1}{x_2}{x_3}$. Find the product $I\xvec$ and explain why $I$ is called the identity matrix.

5. Suppose we write the matrix $A$ in terms of its columns…

1. Consider the linear system Identify the matrix $A$ and vector $\bvec$ to express this system in the form $A\xvec = \bvec$.

2. If $A$ and $\bvec$ are as below, write the linear system corresponding to the equation $A\xvec=\bvec$ and describe its solution space, using a parametric description if appropriate:

3. Describe the solution space of the equation .

4. Suppose $A$ is an $m\times n$ matrix. What can you guarantee about the solution space of the equation $A\xvec = \zerovec$?

Consider the matrices .

1. Before computing, first explain why the shapes of $A$ and $B$ enable us to form the product $AB$. Then describe the shape of $AB$.

2. Compute the product $AB$.

3. Sage can multiply matrices using the * operator. Define the matrices $A$ and $B$ in the Sage cell below and check your work by computing $AB$.

4. Are we able to form the matrix product $BA$? If so, use the Sage cell above to find $BA$. Is it generally…

Consider the system of linear equations .

1. Find the matrix $A$ and vector $\bvec$ that expresses this linear system in the form $A\xvec=\bvec$.

2. Give a description of the solution space to the equation $A\xvec = \bvec$.

Suppose that $A$ is a $135\times2201$ matrix, and that $\xvec$ is a vector. If $A\xvec$ is defined, what is the dimension of $\xvec$? What is the dimension of $A\xvec$?

Suppose that $A$ is a $3\times2$ matrix whose columns are $\vvec_1$ and $\vvec_2$; that is, .

1. What is the dimension of the vectors $\vvec_1$ and $\vvec_2$?

2. What is the product $A\twovec{1}{0}$ in terms of $\vvec_1$ and $\vvec_2$? What is the product $A\twovec{0}{1}$? What is the product $A\twovec{2}{3}$?

3. If we know that what is the matrix $A$?

Suppose that .

1. Describe the solution space to the equation $A\xvec = \zerovec$.

2. Find a $3\times2$ matrix $B$ with no zero entries such that $AB = 0$.

Consider the matrix .

1. Find the product $A\xvec$ where .

2. Give a description of the vectors $\xvec$ such that .

3. Find the reduced row echelon form of $A$ and identify the pivot positions.

4. Can you find a vector $\bvec$ such that $A\xvec=\bvec$ is inconsistent?

5. For a general 3-dimensional vector $\bvec$, what can you say about the solution space of the equation $A\xvec = \bvec$?

The operations that we perform in Gaussian elimination can be accomplished using matrix multiplication. This observation is the basis of an important technique that we will investigate in a subsequent chapter.

Let's consider the matrix .

1. Suppose that . Verify that $SA$ is the matrix that results when the second row of $A$ is scaled by a factor of 7. What matrix $S$ would scale the third row by -3?

2. Suppose t…

In this exercise, you will construct the inverse of a matrix, a subject that we will investigate more fully in the next chapter. Suppose that $A$ is the $2\times2$ matrix: .

1. Find the vectors $\bvec_1$ and $\bvec_2$ such that the matrix $B=\left[\begin{array}{rr} \bvec_1 & \bvec_2 \end{array}\right]$ satisfies .

2. In general, it is not true that $AB = BA$. Check that it is true, however, for the specific $A$ and $B$ that appear in this problem.

3. Suppose that $\xvec = \twovec{x_1}{x_2}$. What do you find when you evaluate $I\xvec$?

4. Suppos…

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

1. If $A\xvec$ is defined, then the number of components of $\xvec$ equals the number of rows of $A$.

2. The solution space to the equation $A\xvec = \bvec$ is equivalent to the solution space to the linear system whose augmented matrix is $\left[\begin{array}{r|r} A & \bvec \end{array}\right]$.

3. If a linear system of equations has 8 equations and 5 unknowns, then the shape of the ma…

Suppose that $A$ is a $4\times4$ matrix and that the equation $A\xvec = \bvec$ has a unique solution for some vector $\bvec$.

1. What does this say about the pivot positions of the matrix $A$? Write the reduced row echelon form of $A$.

2. Can you find another vector $\cvec$ such that $A\xvec = \cvec$ is inconsistent?

3. What can you say about the solution space to the equation $A\xvec = \zerovec$?

4. Suppose $A=\left[\begin{array}{rrrr} \vvec_1 & \vvec_2 & \vvec_3 & \vvec_4 \end{array}\right]$. Explain why every four-dimensional vector can be written as a linear combina…

Define the matrix .

1. Describe the solution space to the homogeneous equation $A\xvec = \zerovec$ using a parametric description, if appropriate. What does this solution space represent geometrically?

2. Describe the solution space to the equation $A\xvec=\bvec$ where $\bvec = \threevec{-3}{-4}{1}$. What does this solution space represent geometrically and how does it compare to the previous solution space?

3. We will now explain the relationship between the previous…

Suppose that a city is starting a bicycle sharing program with bicycles at locations $B$ and $C$. Bicycles that are rented at one location may be returned to either location at the end of the day. Over time, the city finds that 80% of bicycles rented at location $B$ are returned to $B$ with the other 20% returned to $C$. Similarly, 50% of bicycles rented at location $C$ are returned to $B$ and 50% to $C$.

To keep…