Linearity of the Inner Product

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Linearity of the Inner Product

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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 $f(x)$ of a vector $x\ (which we may call an operator on ${\) is said to be linear if for all $x_1\ and $x_2\, and for all scalars $c_1$ and $c_2$ in ${\, we have

\

A linear operator thus “commutes with mixing.”

Linearity consists of two component properties,

  • additivity: $f(x_1+x_2) = f(x_1) + f(x_2)$, and
  • homogeneity: $f(c_1 x_1) = c_1 f(x_1)$.

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.

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