4 
INTRODUCTION, DEFINITIONS 
[chap. I 
we obtain another transformation called the inverse of t 
and denoted by t~ z . It will prove convenient to write 
X, Y for x, y; then 
t~ l 
£ = —X——Y 
K D D ’ 
”=lf*+5 F ' 
Eliminating £ and r] between the four equations defin 
ing t and t~ I , we find that the product tt~ l is 
I: x=X, y = Y, 
which is called the identity transformation I. As would be 
anticipated, also t~ x t=I. 
While t~ I t=tt~ 1 , usually two transformations t and r 
are not commutative, tr^rt, since the sums in (i) are 
usually altered when the Roman and Greek letters are 
interchanged. However, the associative law 
Ctr)T=t{rT) 
holds for any three transformations, so that we may write 
trT without ambiguity. For, if we employ the foregoing 
general transformations t and r, and 
T: X=Au-\-Bv, Y=Cuf-Dv, 
we see that (tr)T is found by eliminating first £, 77 and 
then X, Y between the six equations for t, r, T, while 
t{rT) is obtained by eliminating first X, Y and then 
£, rj between the same equations. Since the same four 
variables are eliminated in each case, we must evidently 
obtain the same final two equations expressing x and y 
in terms of u and v. 
The foregoing definitions and proofs apply at once to 
linear transformations on any number p of variables: