Linear and multilinear algebra; matrix theory Definition

Orthonormal family

Definition

Let $V$ be an inner product space over $\R$ or $\C$ , and let $I$ be a set. A family $(e_i)_{i \in I}$ of vectors of $V$ is orthonormal when, for all $i, j \in I$ , $\langle e_i, e_j \rangle = \delta_{ij},$ where $\delta_{ij} = 1$ if $i = j$ and $\delta_{ij} = 0$ if $i \ne j$ .

Remarks

So each $e_i$ has norm $1$ , and $e_i$ and $e_j$ are orthogonal for $i \ne j$ . The standard basis of $\R^n$ is orthonormal for the dot product. The functions $t \mapsto e^{2\pi i k t}$ , $k \in \Z$ , are orthonormal for $\langle f, g \rangle = \int_0^1 f \bar g\, dt$ .

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Vector space definition
Inner product definition
Orthogonal vectors definition
Orthonormal family definition

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42 pages build on this one.

Tag	`09VYHEF2VT`
URI	`m2://09VYHEF2VT`
Kind	definition
Writing	W0 stub
MSC	15A63, 46C05
Defines	Orthonormal family
Uses	Orthogonal vectors

Orthonormal family

Definition

Remarks

Reading path

Where this leads

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