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
,

*additivity*:*homogeneity*:

The inner product
is *linear in its first argument*, *i.e.*,
for all
, and for all
,

The inner product is also *additive* in its second argument, *i.e.*,

The inner product *is* strictly linear in its second argument with
respect to *real* scalars and :

Since the inner product is linear in both of its arguments for real
scalars, it may be called a *bilinear operator* in that
context.

[How to cite this work] [Order a printed hardcopy]

Copyright ©

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

[Automatic-links disclaimer]