Successive Transformations
Applying multiple transformations
- Original co-ords = \(x\)
- New co-ords = \(y\)
- When I apply matrix \(\mathbf{A}\) to the co-ords \(x\), this can be written as \(\mathbf{A}x=y\)
- If I apply matrix \(\mathbf{A}\) and then matrix \(\mathbf{B}\), then this can be written as \(\mathbf{BA}x=y\)
- You can either multiply out the \(\mathbf{BA}\) matricies first and then do that multiplied by \(x\), or you could do \(\mathbf{B}\times x\) and then that \(\times \mathbf{A}\)
Doing a successive transformation
- \(\begin{pmatrix}2 & 0 \\ 0 & 1\end{pmatrix}\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}\begin{pmatrix}x \\ y\end{pmatrix}=\begin{pmatrix}2 & 0 \\ 0 & 1\end{pmatrix}\begin{pmatrix}y \\ x\end{pmatrix}=\begin{pmatrix}2y \\ x\end{pmatrix}\)
- \(\begin{pmatrix}2 & 0 \\ 0 & 1\end{pmatrix}\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}\begin{pmatrix}x \\ y\end{pmatrix}=\begin{pmatrix}0 & 2 \\ 1 & 0\end{pmatrix}\begin{pmatrix}x \\ y\end{pmatrix}\)
Represent as a single matrix the transformation representing a reflection in the line \(y=x\) followed by a stretch in the x-axis by a factor of 4.
- \(y=x\) matrix \(=\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}\)
- Stretch by 4 in x axis matrix = \(\begin{pmatrix}4 & 0 \\ 0 & 1\end{pmatrix}\)
- \(\begin{pmatrix}4 & 0 \\ 0 & 1\end{pmatrix}\times\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}=\begin{pmatrix}0 & 4 \\ 1 & 0\end{pmatrix}\)
Represent as a single matrix the transformation representing a rotation 90\(^\circ\) anticlockwise about the point (0, 0) followed by a reflection in the line \(y=x\)
- \(y=x\) matrix \(=\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}\)
- Rotation 90\(^\circ\) anticlockwise matrix = \(\begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}\)
- \(\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}\times \begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}=\begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}\)
Exam Question
- P: Matrix for 90\(^\circ\) anticlockwise = \(\begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}\)
- Q: Matrix for the reflection \(y=-x\): \(\begin{pmatrix}0 & -1 \\ -1 & 0\end{pmatrix}\)
- R = QP: \(\begin{pmatrix}0 & -1 \\ -1 & 0\end{pmatrix}\times \begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}=\begin{pmatrix}-1 &0 \\ 0 & 1\end{pmatrix}\)
- Reflection in the y-axis