## 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

Thenand

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

Then and

Adding these two equations givesand subtracting them gives

Substituting these into the original equation of the curvegives

Expanding the brackets giveswhich simplifies to