Plane transformations
Compose operations in their stated order and reverse that order when undoing them
- Column convention The columns of A are the images of the standard basis columns
- Order and centre The last operation is the leftmost factor. Subtract a centre before rotating and add it afterwards
- Check Verify a point, inverse, fixed-point equation and nonnegative area factor as appropriate
Worked example
Reflect in the y-axis, then apply the shear (x,y)↦(x+2y,y)
\[\begin{pmatrix}1&2\\0&1\end{pmatrix}\begin{pmatrix}-1&0\\0&1\end{pmatrix}=\begin{pmatrix}-1&2\\0&1\end{pmatrix}\]
- \(\displaystyle (x,y)\mapsto(-x+2y,y)\)The reflection acts first; verify by substituting an arbitrary point
Common mistakes
Product order
✕First A, then B gives AB✓First A, then B gives BA
BAp means B(Ap)Difficulty levels
introRead and apply a map
standardConstruct and calculate exact transformations
challengeRespect centres and composition order
examInterpret inverse, invariant and area conditions
Questions about this topic
Are fixed points the same as invariant sets?+
A fixed point maps to itself. An invariant set maps onto itself, but its points may move
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, Specialist Mathematics, chapter 7: Trigonometry and transformations
In this chapter
Convert angle measure and use the radian arc relationArea of a sectorUse the sine rule in a non-right-angled triangleUse the cosine rule in a non-right-angled triangleTrigonometry in three dimensionsExact circular-segment area and perimeterChoose perpendicular sections and calculate spatial anglesUse exact trigonometric identities and quadrant informationCompound, double-angle and reciprocal identitiesTransform and reconstruct sine and cosine functionsCombine waves and convert trigonometric sums and productsMap or recover points and finite coordinate setsCompose two plane transformations in orderRotate, reflect and interpret fixed points and area