Mathematik-Online lexicon: Example: Orthogonal Expansion

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	Mathematik-Online lexicon:
	Example: Orthogonal Expansion

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Vector $v=(2,1)^{\operatorname t}$ has the following representation with respect to the orthonormal basis $u_1=\frac{1}{5}(3,4)^{\operatorname t}, u_2=\frac{1}{5}(4,-3)^{\operatorname t}$ :

$\displaystyle v$ $\displaystyle = \langle v, u_1\rangle u_1 + \langle v,u_2\rangle u_2$

$\displaystyle = 2u_1 - u_2\,.$
Vector $v=(-1,1,3)^{\operatorname t}$ has the following representation with respect to the orthonormal basis $u_1=\frac{1}{9}(1,4,8)^{\operatorname t}, u_2=\frac{1}{9}(4,7,-4)^{\operatorname t}, u_3=\frac{1}{9}(8,-4,1)^{\operatorname t}$ :

$\displaystyle v$ $\displaystyle = \langle v, u_1\rangle u_1 + \langle v,u_2\rangle u_2 + \langle v, u_1\rangle u_1$

$\displaystyle = 3u_1 -u_2 -u_3\,.$

(Authors: App/Burkhardt/Höllig)

see also:

automatisch erstellt am 15. 3. 2005

$\displaystyle v$	$\displaystyle = \langle v, u_1\rangle u_1 + \langle v,u_2\rangle u_2$
	$\displaystyle = 2u_1 - u_2\,.$

$\displaystyle v$	$\displaystyle = \langle v, u_1\rangle u_1 + \langle v,u_2\rangle u_2 + \langle v, u_1\rangle u_1$
	$\displaystyle = 3u_1 -u_2 -u_3\,.$