6.2. Composite linear transformations#

Definition 6.5 (Composite transformations)

Let \(T : V \to W\) and \(S: W \to X\) be two linear transformations over the vector spaces \(V, W\) and \(X\). The composition of \(S\) and \(T\) is the transformation \(S \circ T: V \to X\) defined by

\[ (S \circ T)(\mathbf{u}) = S(T(\mathbf{u})), \]

for all vectors \(\mathbf{u} \in V\).

For example, two linear transformations are defined as \(T:(x, y, z) \mapsto (2 x + 4 y, -x + 3 y, x + 2 y)\) and \(S:(x, y, z) \mapsto (2x + y - z, 3x + z, y - 2z)\). The composite linear transformation \(S \circ T(x, y, z)\) is

\[\begin{split} \begin{align*} S \circ T\begin{pmatrix} x \\ y \\ z \end{pmatrix} &= S \left( T\begin{pmatrix} x \\ y \\ z \end{pmatrix} \right) = S \begin{pmatrix} 2 x + 4 \\ - x + 3y \\ x + 2 y \end{pmatrix} \\ &= \begin{pmatrix} 2(2 x + 4 y) + (-x + 3 y) - (x + 2 y) \\ 3(2 x + 4 y) + (x + 2 y) \\ (-x + 3 y) - 2(x + 2 y) \end{pmatrix} \\ &= \begin{pmatrix} 2 x + 9 y \\ 7 x + 14 y \\ -3 x - y \end{pmatrix}, \end{align*} \end{split}\]

i.e., \(S \circ T(x,y , z) \mapsto (2x + 9y, 7x + 14y, -3x - y)\).

6.2.1. Composite transformation matrices#

We have seen that a linear transformation \(T: V \to W\) can be represented by a transformation matrix so that given a vector \(\mathbf{u} \in V\) the image is calculated using

(6.3)#\[ T(\mathbf{u}) = A \mathbf{u}. \]

Consider the composition of \(T\) with another linear transformation \(S: W \to X\) with a transformation matrix \(B\)

(6.4)#\[ S \circ T(\mathbf{u}) = B \cdot T(\mathbf{u}) \]

Substituting equation (6.3) into equation (6.3) gives

\[ S \circ T (\mathbf{u}) = B \cdot A \mathbf{u}. \]

Theorem 6.2 (Composite transformation matrices)

Given two linear transformations \(T:V \to W\) and \(S:W \to X\) with transformation matrices \(A\) and \(B\) respectively then the composition \(S \circ T\) of the vector \(\mathbf{u} \in V\) is[1]

(6.5)#\[ S \circ T (\mathbf{u}) = B \cdot A \cdot \mathbf{u}. \]

Example 6.5

Calculate the transformation matrices, \(A\) and \(B\), for the transformations \(T:(x, y, z) \mapsto (2 x + 4 y, -x + 3 y, x + 2 y)\) and \(S:(x, y, z) \mapsto (2x + y - z, 3x + z, y - 2z)\). Use them to calculate the transformation matrix \(C\) for \(S\circ T\).

Solution

The transformation matrices for \(T\) and \(S\) are

\[\begin{split} \begin{align*} A &= \begin{pmatrix} 2 & 4 \\ -1 & 3 \\ 1 & 2 \end{pmatrix}, & B &= \begin{pmatrix} 2 & 1 & - 1 \\ 3 & 0 & 1 \\ 0 & 1 & -2 \end{pmatrix}. \end{align*} \end{split}\]

The transformation matrix for the composite transformation \(S \circ T\) is \(C = BA\)

\[\begin{split} \begin{align*} C &= BA = \begin{pmatrix} 2 & 1 & - 1 \\ 3 & 0 & 1 \\ 0 & 1 & -2 \end{pmatrix} \begin{pmatrix} 2 & 4 \\ -1 & 3 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} 2 & 9 \\ 7 & 14 \\ -3 & -1 \end{pmatrix}. \end{align*} \end{split}\]

So \(S \circ T(x,y , z) \mapsto (2x + 9y, 7x + 14y, -3x - y)\) which we saw in the example above