Notes
- Suppose we combine two linear transformations, first reflecting in the x-axis, then rotating 90 degrees clockwise about the origin. Since we can represent each transformation with a matrix, we simply multiply the matrices together to find the matrix that represents the combined transformation.
- First, reflect in the x-axis: [latex]A\,\mathbf{x}=\left(\begin{array}{c}1 & 0\\0 & -1\end{array}\right)\left(\begin{array}{c}x\\y\end{array}\right)=\left(\begin{array}{c}x\\-y\end{array}\right)[/latex].
- Next, rotate 90 degrees clockwise about the origin: [latex]DA\,\mathbf{x}=\left(\begin{array}{c}0 & 1\\-1 & 0\end{array}\right)\left(\begin{array}{c}1 & 0\\0 & -1\end{array}\right)\left(\begin{array}{c}x\\y\end{array}\right)[/latex][latex]=\left(\begin{array}{c}0 & 1\\-1 & 0\end{array}\right)\left(\begin{array}{c}x\\-y\end{array}\right)=\left(\begin{array}{c}-y\\-x\end{array}\right)[/latex].
- The matrix that represents the combined transformation is therefore: [latex]DA\,\mathbf{x}=\left(\begin{array}{c}0 & 1\\-1 & 0\end{array}\right)\left(\begin{array}{c}1 & 0\\0 & -1\end{array}\right)\left(\begin{array}{c}x\\y\end{array}\right)[/latex][latex]=\left(\begin{array}{c}0 & -1\\-1 & 0\end{array}\right)\left(\begin{array}{c}x\\y\end{array}\right)=\left(\begin{array}{c}-y\\-x\end{array}\right)[/latex].
- Note that the matrices are multiplied in the opposite order to the transformations because we’re applying the second transformation to the result of the first transformation.
Video Tips
Practice Exercises
Drag and drop the matrices below to their correct locations. Some matrices fit in more than one place and some don’t fit anywhere.
