6.4. 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

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

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

Example 6.5

Two linear transformations is defined as \(T:(x, y, z)^\mathsf{T} \mapsto (2 x + 4 y, -x + 3 y, x + 2 y)^\mathsf{T}\) and \(S:(x, y, z)^\mathsf{T} \mapsto (2x + y - z, 3x + z, y - 2z)^\mathsf{T}\), determine the composite linear transformation \(S \circ T(\vec{u})\) for \(\vec{u} = (x, y, z)^\mathsf{T}\).

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

6.4.1. 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 \(\vec{u} \in V\) the image is calculated using

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

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

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

Substituting equation (6.3) into equation (6.3) gives

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

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 \(\vec{u} \in V\) is[1]

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

Example 6.6

Calculate the transformation matrices, \(A\) and \(B\), for the transformations \(T\) and \(S\) from Example 6.5 and use them to calculate the transformation matrix \(C\) for \(S\circ T\).

Solution (click to show)

The transformations are \(T:(x, y, z)^\mathsf{T} \mapsto (2 x + 4 y, -x + 3 y, x + 2 y)^\mathsf{T}\) and \(S:(x, y, z)^\mathsf{T} \mapsto (2x + y - z, 3x + z, y - 2z)^\mathsf{T}\) which have the transformation matrices

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

Calculate the composite transformation matrix for \(S \circ T\)

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

Checking that \(C \vec{u}\) gives the answer from Example 6.5

\[\begin{split} \begin{align*} C \cdot \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 2 & 9 \\ 7 & 14 \\ -3 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 2 x + 9 y \\ 7 x + 14 y \\ -3 x + y \end{pmatrix} \quad \checkmark \end{align*} \end{split}\]