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Linearity of the Inner Product
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:
- additivity:
- homogeneity:
A function of multiple vectors,
e.g.,

can be linear or not
with respect to each of its arguments.
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.
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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at
Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See
http://ccrma.stanford.edu/~jos/ for details.