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 $(S \circ T)(\vec{v})=S(T(\vec{v}))\text{.}$

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 $\IR^3$

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. $\displaystyle 2 \times 4$

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 $A=\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ -1 & -2 \end{array}\right]\text{.}$

Part 1.

Compute

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 $A=\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ -1 & -2 \end{array}\right]\text{.}$

Part 2.

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

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 $A=\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ -1 & -2 \end{array}\right]\text{.}$

Part 3.

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

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 $A=\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ -1 & -2 \end{array}\right]\text{.}$

Part 4.

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

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:

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 $B=\left[\begin{array}{cc} 2 & 3 \\ 1 & -1 \\ 0 & -1 \end{array}\right]\text{.}$

Part 1.

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

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 $B=\left[\begin{array}{cc} 2 & 3 \\ 1 & -1 \\ 0 & -1 \end{array}\right]\text{.}$

Part 2.

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

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 $B=\left[\begin{array}{cc} 2 & 3 \\ 1 & -1 \\ 0 & -1 \end{array}\right]\text{.}$

Part 3.

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

Consider the following three matrices.

Part 1.

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

Consider the following three matrices.

Part 2.

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

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

Part 1.

Compute the product $BA$ by hand.

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

Part 2.

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
    

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

Answer.