Matrices of Composite Transformations (DP IB Applications & Interpretation (AI): HL): Revision Note

Matrices of composite transformations

What is a composite transformation?

• A composite transformation is the result of applying more than one transformation to an object

  • e.g. a rotation followed by an enlargement

• The order of transformations is important

  • e.g. a rotation followed by a reflection can be different to the reflection followed by the rotation

• It is possible to find a single composite function matrix that does the same job as applying the individual transformation matrices

How do I find a single matrix representing a composite transformation?

• You use matrix multiplication to find the single matrix of a composite transformation

• The single matrix is $\mathit{S}\mathit{T}$ where

  • $\mathit{T}$ is the matrix for the first transformation

  • $\mathit{S}$ is the matrix for the second transformation

• You can extend this to multiple transformations

  • Start with the first matrix

  • Then pre-multiply by the next matrix

  • And so on

How do I apply the same transformation matrix more than once?

• The matrix for the transformation $\mathit{T}$ applied twice is ${\mathit{T}}^2$

• The matrix for the transformation $\mathit{T}$ applied $n$times is ${\mathit{T}}^n$

• You can use your knowledge of transformations to form the identity matrix

  • e.g. if $\mathit{R}$ is the matrix for a rotation of 30° clockwise then ${\mathit{R}}^{12}=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$

  • e.g. if $\mathit{S}$ is the matrix for a reflection in the line $y=mx$ then ${\mathit{S}}^2=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$