**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION:** "Mathematics of the Discrete
Fourier Transform (DFT), with Audio Applications --- Second
Edition", by Julius
O. Smith III, W3K
Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright ©
*2017-09-28* by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University

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## Linearity of the Inner Product

Any function of a vector (which we may call an

operatoron ) is said to belinearif for all and , and for all scalars and in , we have

A linear operator thus ''commutes with mixing.''Linearity consists of two component properties,

additivity: , andhomogeneity: .The inner product is

linearin its first argument, i.e.

This is easy to show from the definition:

The inner product is also

additivein its second argument, i.e.,

but it is onlyconjugate homogeneousin its second argument, since

The inner product

isstrictly linear in its second argument with respect torealscalars:

Since the inner product is linear in both of its arguments for real scalars, it is often called a

bilinear operatorin that context.