Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

PART II*

FINITE GROUPS OF LINEAR HOMOGENEOUS TRANSFORMATIONS
CHAPTER IX
PRELIMINARY THEOREMS
LINEAR TRANSFORMATIONS, ~~ 75-82
75. Introduction and Definition. It is often of importance
in analysis to exchange one set of variables for another, the
variables of either set being linear homogeneous functions of
the variables of the other set (cf. Ch. XVIII), as in coordinate
geometry:
x=x' cos 0-y' sin 0,
(1)
y=x' sin 0+y' cos 0.
We assume that a function f(x, y) is given, in which the new
variables (x', y') are to be put in place of the old (x, y) by means
of (1); this is called operating upon f by the linear transformation (1).
A capital letter is in general used to denote a linear transformation; thus, we shall here denote (1) by S. The result
of operating upon f(x, y) by S may then be indicated symbolically as follows:
(2) (f)S=f(x' cos 0-y' sin 0, x' sin O+y' cos 0).
* This part was written by H. F. Blichfeldt.
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