If \(T: \IR^n \rightarrow \IR^m\) and \(S: \IR^m \rightarrow \IR^k\) are linear maps, then the composition map \(S\circ T\) computed as \((S \circ T)(\vec{v})=S(T(\vec{v}))\) is a linear map from \(\IR^n \rightarrow \IR^k\text{.}\)

Let \(T: \IR^3 \rightarrow \IR^2\) be defined by the \(2\times 3\) starndard matrix \(B\) and \(S: \IR^2 \rightarrow \IR^4\) be defined by the \(4\times 2\) standard matrix \(A\text{:}\)

Part 1.

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 defined by the \(2\times 3\) starndard matrix \(B\) and \(S: \IR^2 \rightarrow \IR^4\) be defined by the \(4\times 2\) standard matrix \(A\text{:}\)

Part 2.

What size will the standard matrix of \(S \circ T\) be?

1. \(\displaystyle 4 \text{ (rows)} \times 3 \text{ (columns)}\)
2. \(\displaystyle 3 \text{ (rows)} \times 4 \text{ (columns)}\)
3. \(\displaystyle 3 \text{ (rows)} \times 2 \text{ (columns)}\)
4. \(\displaystyle 2 \text{ (rows)} \times 4 \text{ (columns)}\)

Let \(T: \IR^3 \rightarrow \IR^2\) be defined by the \(2\times 3\) starndard matrix \(B\) and \(S: \IR^2 \rightarrow \IR^4\) be defined by the \(4\times 2\) standard matrix \(A\text{:}\)

Part 3.

Compute

Let \(T: \IR^3 \rightarrow \IR^2\) be defined by the \(2\times 3\) starndard matrix \(B\) and \(S: \IR^2 \rightarrow \IR^4\) be defined by the \(4\times 2\) standard matrix \(A\text{:}\)

Part 4.

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

Let \(T: \IR^3 \rightarrow \IR^2\) be defined by the \(2\times 3\) starndard matrix \(B\) and \(S: \IR^2 \rightarrow \IR^4\) be defined by the \(4\times 2\) standard matrix \(A\text{:}\)

Part 5.

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

Let \(T: \IR^3 \rightarrow \IR^2\) be defined by the \(2\times 3\) starndard matrix \(B\) and \(S: \IR^2 \rightarrow \IR^4\) be defined by the \(4\times 2\) standard matrix \(A\text{:}\)

Part 6.

Use \((S \circ T)(\vec{e}_1),(S \circ T)(\vec{e}_2),(S \circ T)(\vec{e}_3)\) to write the standard matrix for \(S \circ T\text{.}\)

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.