Suppose we have a curve which undergoes a linear transformation. The transformation may be represented by a matrixand 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 lineafter transformation by the matrix
write line line as the vector
then
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 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 curveis rotated by
The matrix representing this rotation is
and
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