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*:

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

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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University