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Mathematics-Online course: Linear Algebra - Basic Structures - Bases

Orthogonal Basis


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A given basis $ B = \{u_1,\dots,u_n\}$ is called orthogonal if

$\displaystyle \langle u_i,u_j \rangle = 0,\quad i \neq j\,
.
$

If all basis vectors are unit vectors, that is $ \vert u_i\vert = 1$, then $ B$ is called orthonormal system or orthonormal basis.

A vector $ v$ has the following representation relative to a given orthogonal basis $ u_1,\ldots,u_n$:

$\displaystyle v = \sum_{j=1}^n
\frac{\langle v,u_j\rangle}{\langle u_j,u_j\rangle}\,
u_j\,
.
$

For the coefficients

$\displaystyle c_j = \frac{\langle v,u_j\rangle}{\vert u_j\vert^2}\,
,
$

we have

$\displaystyle \vert c_1\vert^2 \vert u_1\vert^2 + \cdots + \vert c_n\vert^2 \vert u_n\vert^2
= \vert v\vert^2\,
.
$

For a orthonormal basis the denominators equal one and this leads to

$\displaystyle c_j = \langle v,u_j\rangle\,,\quad
\vert c_1\vert^2 + \cdots + \vert c_n\vert^2 = \vert v\vert^2\,.
$

(Authors: App/Burkhardt/Höllig/Kimmerle)

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  automatically generated 4/21/2005