## 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 matrix and the curve by a vector where may be a function of or 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 line after transformation by the matrix write line line as the vector then Then and Make the subject of both equations and equate the result to give Now make y' the subject to give Finally drop the ' to give More generally we multiply the matrix by the vector obtaining and in terms of and then solve these equations to find and in terms of and Finally substitute for and into the original equation of the curve to obtain an equation relating and Finally drop the ' as in the example above.

Suppose that the curve is rotated by The matrix representing this rotation is and Then and Adding these two equations gives and subtracting them gives Substituting these into the original equation of the curve gives Expanding the brackets gives which simplifies to  