Dilations
LearningObjectives
- Use the languageof dilations.
- Calculateand applyscalarproducts.
- Use scalarproductsto representdilations.
Introduction
We beginthe lessonwith a reviewof dilations,whichwereintroducedin an earlierchapter. Like the other
transformations,dilationscan be expressedusingmatrices.Beforewe can do that, though,you will learn
abouta secondkind of multiplicationwith matricescalledscalarmultiplication.
DilationRefresher
The imageof point in a dilationcenteredat the origin,with a scalefactor , is the point.
For , the dilationis an enlargement.
For , the dilationis a reduction.
Any linearfeatureof animageis timesas long as the lengthin the originalfigure.
Areasin theimageare timesthe correspondingarea in the originalfigure.
ScalarMultiplication
In an earlierlessonyou learnedaboutmatrixmultiplication:multiplicationof one matrixby anothermatrix.
Scalarmultiplicationis the multiplicationof a matrixby a real number. The productin scalarmultiplication
is a matrix.Eachelementof the originalmatrixis multipliedby the scalar(the real number)to producethe
correspondingelementin the scalarproduct.