Linearity of the Inner Product

Any function $f(\underline{u})$ of a vector $\underline{u}\in{\bf C}^N$ (which we may call an operator on ${\bf C}^N$ ) is said to be linear if for all $\underline{u}\in{\bf C}^N$ and $\underline{v}\in{\bf C}^N$ , and for all scalars $\alpha$ and $\beta$ in ${\bf C}$ ,

$\displaystyle f(\alpha \underline{u}+ \beta \underline{v}) = \alpha f(\underline{u}) + \beta f(\underline{v}).$

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

additivity: $f(\underline{u}+\underline{v}) = f(\underline{u}) + f(\underline{v})$
homogeneity: $f(\alpha \underline{u}) = \alpha f(\underline{u})$

A function of multiple vectors, e.g., $f(\underline{u},\underline{v},\underline{w})$ can be linear or not with respect to each of its arguments.

The inner product $\left<\underline{u},\underline{v}\right>$ is linear in its first argument, i.e., for all $\underline{u},\underline{v},\underline{w}\in{\bf C}^N$ , and for all $\alpha, \beta\in{\bf C}^N$ ,

$\displaystyle \left<\alpha \underline{u}+ \beta \underline{v},\underline{w}\rig... ...line{u},\underline{w}\right> + \beta \left<\underline{v},\underline{w}\right>.$

This is easy to show from the definition:

$\begin{eqnarray*} \left<\alpha \underline{u}+ \beta \underline{v},\underline{w}\... ...rline{w}\right> + \beta \left<\underline{v},\underline{w}\right> \end{eqnarray*}$

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

$\displaystyle \left<\underline{u},\underline{v}+ \underline{w}\right> = \left<\underline{u},\underline{v}\right> + \left<\underline{u},\underline{w}\right>,$

but it is only conjugate homogeneous (or antilinear) in its second argument, since

$\displaystyle \left<\underline{u},\alpha \underline{v}\right> = \overline{\alph... ...{u},\underline{v}\right> \neq \alpha \left<\underline{u},\underline{v}\right>.$

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

$\displaystyle \left<\underline{u},a \underline{v}+ b \underline{w}\right> = a \... ...ne{v}\right> + b \left<\underline{u},\underline{w}\right>, \quad a,b\in{\bf R}$

where $\underline{u},\underline{v},\underline{w}\in{\bf C}^N$ .

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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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Mathematics of the DFT
    Geometric Signal Theory
       The Inner Product
          Linearity of the Inner Product