Any function
of a vector
(which we may call an
operator on
) is said to be linear if for all
and
, and for all scalars
and
in
,
A linear operator thus ``commutes with mixing.'' Linearity consists of two component properties:
The inner product
is linear in its first argument, i.e.,
for all
, and for all
,
This is easy to show from the definition:
The inner product is also additive in its second argument, i.e.,
but it is only conjugate homogeneous (or antilinear) in its second argument, since
The inner product is strictly linear in its second argument with
respect to real scalars
and
:
where
Since the inner product is linear in both of its arguments for real scalars, it may be called a bilinear operator in that context.