Section 4.1 Matrices and Multiplication (M1)
Observation 4.1.1.
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 \((S \circ T)(\vec{v})=S(T(\vec{v}))\text{.}\)
Activity 4.1.3.
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 are the domain and codomain of the composition map \(S \circ T\text{?}\)
The domain is \(\IR ^2\) and the codomain is \(\IR^3\)
The domain is \(\IR ^3\) and the codomain is \(\IR^2\)
The domain is \(\IR ^2\) and the codomain is \(\IR^4\)
The domain is \(\IR ^3\) and the codomain is \(\IR^4\)
The domain is \(\IR ^4\) and the codomain is \(\IR^3\)
The domain is \(\IR ^4\) and the codomain is \(\IR^2\)
Activity 4.1.4.
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 \(A=\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ -1 & -2 \end{array}\right]\text{.}\)
What size will the standard matrix of \(S \circ T:\IR^3\to\IR^4\) be? (Rows \(\times\) Columns)
\(\displaystyle 4 \times 3\)
\(\displaystyle 4 \times 2\)
\(\displaystyle 3 \times 4\)
\(\displaystyle 3 \times 2\)
\(\displaystyle 2 \times 4\)
\(\displaystyle 2 \times 3\)
Activity 4.1.5.
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
(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 \(S \circ T:\IR^3\to\IR^4\text{.}\)
Definition 4.1.6.
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:
Activity 4.1.7.
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 \(T \circ S\text{.}\)
Activity 4.1.8.
Consider the following three matrices.
(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.
Activity 4.1.9.
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
Subsection 4.1.1 Videos
Exercises 4.1.2 Exercises
Exercises available at checkit.clontz.org 1 .
https://checkit.clontz.org/#/banks/tbil-la/M1/