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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\,. $

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  automatisch erstellt am 19.  8. 2013