Finding the Equation of a Curve After Transformation by a Matrix
Suppose we have a curve which undergoes a linear transformation. The transformation may be represented by a matrixand the curve by a vectorwheremay be a function ofor vice versa, or both are functions of some parameter (I will not deal with this case here).
The simplest case is when a line is transformed. To find the equation of the lineafter transformation by the matrixwrite line line as the vectorthen
Makethe subject of both equations and equate the result to give
Now make y' the subject to giveFinally drop the ' to give
More generally we multiply the matrixby the vectorobtainingandin terms ofandthen solve these equations to findandin terms ofandFinally substitute forandinto the original equation of the curve to obtain an equation relating andFinally drop the ' as in the example above.
Suppose that the curveis rotated byThe matrix representing this rotation isand
Adding these two equations givesand subtracting them gives
Substituting these into the original equation of the curvegives
Expanding the brackets giveswhich simplifies to