Section4.1Matrices and Multiplication (MX1)

Subsection4.1.1Class Activities

Observation4.1.1.

If $$T: \IR^n \rightarrow \IR^m$$ and $$S: \IR^m \rightarrow \IR^k$$ are linear maps, then the composition map $$S\circ T$$ is a linear map from $$\IR^n \rightarrow \IR^k\text{.}$$

Recall that for a vector, $$\vec{v} \in \IR^n\text{,}$$ the composition is computed as $$(S \circ T)(\vec{v})=S(T(\vec{v}))\text{.}$$

Activity4.1.2.

Let $$T: \IR^3 \rightarrow \IR^2$$ be given by the $$2\times 3$$ standard matrix $$B=\left[\begin{array}{ccc} 2 & 1 & -3 \\ 5 & -3 & 4 \end{array}\right]$$ and $$S: \IR^2 \rightarrow \IR^4$$ be given by the $$4\times 2$$ standard matrix $$A=\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ -1 & -2 \end{array}\right]\text{.}$$

What are the domain and codomain of the composition map $$S \circ T\text{?}$$

1. The domain is $$\IR ^3$$ and the codomain is $$\IR^2$$

2. The domain is $$\IR ^2$$ and the codomain is $$\IR^4$$

3. The domain is $$\IR ^3$$ and the codomain is $$\IR^4$$

4. The domain is $$\IR ^4$$ and the codomain is $$\IR^3$$

Activity4.1.3.

Let $$T: \IR^3 \rightarrow \IR^2$$ be given by the $$2\times 3$$ standard matrix $$B=\left[\begin{array}{ccc} 2 & 1 & -3 \\ 5 & -3 & 4 \end{array}\right]$$ and $$S: \IR^2 \rightarrow \IR^4$$ be given by the $$4\times 2$$ standard matrix $$A=\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ -1 & -2 \end{array}\right]\text{.}$$

What size will the standard matrix of $$S \circ T:\IR^3\to\IR^4$$ be? (Rows $$\times$$ Columns)

1. $$\displaystyle 4 \times 3$$

2. $$\displaystyle 3 \times 4$$

3. $$\displaystyle 3 \times 2$$

4. $$\displaystyle 2 \times 4$$

Activity4.1.4.

Let $$T: \IR^3 \rightarrow \IR^2$$ be given by the $$2\times 3$$ standard matrix $$B=\left[\begin{array}{ccc} 2 & 1 & -3 \\ 5 & -3 & 4 \end{array}\right]$$ and $$S: \IR^2 \rightarrow \IR^4$$ be given by the $$4\times 2$$ standard matrix $$A=\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ -1 & -2 \end{array}\right]\text{.}$$

(a)

Compute

\begin{equation*} (S \circ T)(\vec{e}_1) = S(T(\vec{e}_1)) = S\left(\left[\begin{array}{c} 2 \\ 5\end{array}\right]\right) = \left[\begin{array}{c}\unknown\\\unknown\\\unknown\\\unknown\end{array}\right]. \end{equation*}
(b)

Compute $$(S \circ T)(\vec{e}_2) \text{.}$$

(c)

Compute $$(S \circ T)(\vec{e}_3) \text{.}$$

(d)

Write the $$4\times 3$$ standard matrix of $$S \circ T:\IR^3\to\IR^4\text{.}$$

Definition4.1.5.

We define the product $$AB$$ of a $$m \times n$$ matrix $$A$$ and a $$n \times k$$ matrix $$B$$ to be the $$m \times k$$ standard matrix of the composition map of the two corresponding linear functions.

For the previous activity, $$T$$ was a map $$\IR^3 \rightarrow \IR^2\text{,}$$ and $$S$$ was a map $$\IR^2 \rightarrow \IR^4\text{,}$$ so $$S \circ T$$ gave a map $$\IR^3 \rightarrow \IR^4$$ with a $$4\times 3$$ standard matrix:

\begin{equation*} AB = \left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ -1 & -2 \end{array}\right] \left[\begin{array}{ccc} 2 & 1 & -3 \\ 5 & -3 & 4 \end{array}\right] \end{equation*}
\begin{equation*} = \left[ (S \circ T)(\vec{e}_1) \hspace{1em} (S\circ T)(\vec{e}_2) \hspace{1em} (S \circ T)(\vec{e}_3) \right] = \left[\begin{array}{ccc} 12 & -5 & 5 \\ 5 & -3 & 4 \\ 31 & -12 & 11 \\ -12 & 5 & -5 \end{array}\right] . \end{equation*}

Activity4.1.6.

Let $$S: \IR^3 \rightarrow \IR^2$$ be given by the matrix $$A=\left[\begin{array}{ccc} -4 & -2 & 3 \\ 0 & 1 & 1 \end{array}\right]$$ and $$T: \IR^2 \rightarrow \IR^3$$ be given by the matrix $$B=\left[\begin{array}{cc} 2 & 3 \\ 1 & -1 \\ 0 & -1 \end{array}\right]\text{.}$$

(a)

Write the dimensions (rows $$\times$$ columns) for $$A\text{,}$$ $$B\text{,}$$ $$AB\text{,}$$ and $$BA\text{.}$$

(b)

Find the standard matrix $$AB$$ of $$S \circ T\text{.}$$

(c)

Find the standard matrix $$BA$$ of $$T \circ S\text{.}$$

Activity4.1.7.

Consider the following three matrices.

\begin{equation*} A = \left[\begin{array}{ccc}1&0&-3\\3&2&1\end{array}\right] \hspace{2em} B = \left[\begin{array}{ccccc}2&2&1&0&1\\1&1&1&-1&0\\0&0&3&2&1\\-1&5&7&2&1\end{array}\right] \hspace{2em} C = \left[\begin{array}{cc}2&2\\0&-1\\3&1\\4&0\end{array}\right] \end{equation*}
(a)

Find the domain and codomain of each of the three linear maps corresponding to $$A\text{,}$$ $$B\text{,}$$ and $$C\text{.}$$

(b)

Only one of the matrix products $$AB,AC,BA,BC,CA,CB$$ can actually be computed. Compute it.

Activity4.1.8.

Let $$B=\left[\begin{array}{ccc} 3 & -4 & 0 \\ 2 & 0 & -1 \\ 0 & -3 & 3 \end{array}\right]\text{,}$$ and let $$A=\left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right]\text{.}$$

(a)

Compute the product $$BA$$ by hand.

(b)

Check your work using technology. Using Octave:

B = [3 -4 0 ; 2 0 -1 ; 0 -3 3]
A = [2 7 -1 ; 0 3 2  ; 1 1 -1]
B*A

B = [3 -4 0 ; 2 0 -1 ; 0 -3 3]
A = [2 7 -1 ; 0 3 2  ; 1 1 -1]
B*A


Activity4.1.9.

Of the following three matrices, only two may be multiplied.

\begin{equation*} A=\left[\begin{array}{cccc} -1 & 3 & -2 & -3 \\ 1 & -4 & 2 & 3 \end{array}\right] \hspace{1em} B=\left[\begin{array}{ccc} 1 & -6 & -1 \\ 0 & 1 & 0 \end{array}\right] \hspace{1em} C=\left[\begin{array}{ccc} 1 & -1 & -1 \\ 0 & 1 & -2 \\ -2 & 4 & -1 \\ -2 & 3 & -1 \end{array}\right] \end{equation*}

Explain which two can be multiplied and why. Then show how to find their product.

\begin{equation*} AC=\left[\begin{array}{ccc} 9 & -13 & 0 \\ -9 & 12 & 2 \end{array}\right] \end{equation*}

Subsection4.1.3Slideshow

Slideshow of activities available at https://teambasedinquirylearning.github.io/linear-algebra/2022/MX1.slides.html.

Subsection4.1.5Mathematical Writing Explorations

Exploration4.1.10.

Construct 3 matrices, $$A,B,\mbox{ and } C\text{,}$$ such that
• $$\displaystyle AB:\mathbb{R}^4\rightarrow\mathbb{R}^2$$

• $$\displaystyle BC:\mathbb{R}^2\rightarrow\mathbb{R}^3$$

• $$\displaystyle CA:\mathbb{R}^3\rightarrow\mathbb{R}^4$$

• $$\displaystyle ABC:\mathbb{R}^2\rightarrow\mathbb{R}^2$$

Exploration4.1.11.

Construct 3 examples of matrix multiplication, with all matrix dimensions at least 2.
• Where $$A$$ and $$B$$ are not square, but $$AB$$ is square.

• Where $$AB = BA\text{.}$$

• Where $$AB \neq BA\text{.}$$

Exploration4.1.12.

Use the included map in this problem.
• An adjacency matrix for this map is a matrix that has the number of roads from city $$i$$ to city $$j$$ in the $$(i,j)$$ entry of the matrix. A road is a path of length exactly 1. All $$(i,i)$$entries are 0. Write the adjacency matrix for this map, with the cities in alphabetical order.

• What does the square of this matrix tell you about the map? The cube? The $$n$$-th power?

Subsection4.1.6Sample Problem and Solution

Sample problem Example B.1.18.