NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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 operator on ) is said to be linear if 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: , and
- homogeneity: .
The inner product is linear in its first argument, i.e.
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 in its second argument, since
The inner product is strictly linear in its second argument with respect to real scalars:
Since the inner product is linear in both of its arguments for real scalars, it is often called a bilinear operator in that context.