5RN43XCT

[ Close ] Wikipedia's W.svg Warmly celebrate the number of entries in the Chinese Wikipedia exceeded one million ! The Hsinchu Wikimedia Community Writing Meeting is held on the first weekend of each month. Registration is welcome. [ Close ] Standard orthogonal base Wikipedia, the free encyclopedia In linear algebra , one inner product space of orthogonal basis ( Orthogonal Basis ) twenty-two element is perpendicular to the base . The base element is called the base vector . If the norm of a base vector of an orthonormal basis is unit length 1, then the orthonormal basis or Orthonormal basis is referred to as the orthonormal basis . The concept of orthogonal basis is very important in both finite and infinite dimensions. In the infinite-dimensional Hilbert space , the orthogonal group is no longer Hamoerji , is to say each element can not be written as a linear combination of a finite number of groups of elements. Therefore, in an infinite-dimensional space, the orthogonal base should be more strictly defined as an element that is linearly independent and bi-orthogonal, and the spanned space is a set of dense subspaces (rather than the entire space) of the original space . Note that in the space where no inner product is defined, the term "orthogonal basis" is meaningless. Therefore, a Banach space with orthogonal basis is a Hilbert space . table of Contents 1 example 2 basic properties 3 Existence of orthogonal groups 4 Hamelki 5 See Example In Euclidean space {\displaystyle \mathbb {R} ^{3}} {\mathbb {R}}^{{3}}In the set, { e 1 =(1,0,0), e 2 =(0,1,0), e 3 =(0,0,1)} form an orthonormal basis. A set defined by f n ( x ) = exp(2π inx ): { f n : n ∈ Z } constitutes an orthonormal basis on the Fourier space L 2 ([0,1]). Basic properties B is H an orthogonal basis on, then H each element of x can be expressed as: {\displaystyle x=\sum _{b\in B}{\langle x,b\rangle \over \lVert b\rVert ^{2}}b} x=\sum _{{b\in B}}{\langle x,b\rangle \over \lVert b\rVert ^{2}}b When B is a standard orthogonal basis, it is: {\displaystyle x=\sum _{b\in B}\langle x,b\rangle b} x=\sum _{{b\in B}}\langle x,b\rangle b The module length of x is expressed as: {\displaystyle \|x\|^{2}=\sum _{b\in B}|\langle x,b\rangle |^{2}} \|x\|^{2}=\sum _{{b\in B}}|\langle x,b\rangle |^{2}. Even if B is not countable, the non-zero entries in the above formula are only countable, so this expression is still valid. The above formula is called the Fourier expansion of x , see the Fourier series . If B is H an orthonormal basis on, then H " isomorphic " in sequence space L 2 ( B ). Because there H -> L 2 ( B ) is a bijection Φ, such that all H of x and y are: {\displaystyle \langle \Phi (x),\Phi (y)\rangle =\langle x,y\rangle } \langle \Phi (x),\Phi (y)\rangle =\langle x,y\rangle The existence of orthogonal bases Using Zorn's lemma and Gram-Schmidt orthogonalization method, it can be proved that every Hilbert space has a base and an orthogonal base. The cardinality of orthogonal bases in the same space must be the same. When a Hilbert space has an orthonormal base consisting of countable elements, this space is said to be separable. Hamelji With the previous definition, it can be known that in the infinite dimension space, the orthonormal basis is no longer the base of the definition of general linear algebra. For distinction, the base under the definition of a general linear algebra is called Hamelji. In practical applications of the inner product space, Hamelji rarely appears, so when referring to the concept of "base," it is generally referred to as orthogonal basis. See also Base (linear algebra) Orthogonal Orthogonalization Gram-Schmidt Orthogonalization Orthogonal decomposition Orthogonal matrix vertical 2 categories :Abstract algebraLinear algebra Navigation menu No logindialoguecontributionCreate an accountSign inentrydiscuss Taiwanese readeditView historysearch 搜尋維基百科 Home Classification index Featured content news Recent changes Random entry Instructions Instructions Wiki community Policies and Guidelines Mutual aid Quiz Word conversion IRC Live Chat contact us About Wikipedia Funding Wikipedia Print/Export Download as PDF Printable version tool Links to this page Related changes Upload file Special page Static link Page information Wikidata project Quote this page Left side jumper link other languages العربية Deutsch English Español Français Japanese 한국어 Português Русский There are 14 languages Edit link This page was last modified on March 17, 2015 (Tuesday) at 03:11. All texts on this site are provided under the terms of the Creative Commons Attribution-Share Alike 3.0 Agreement , and additional terms may also apply (see Terms of Use ). Wikipedia® and Wikipedia logo is the Wikimedia Foundation, a registered trade mark; Wikipedia ™ is a trademark of the Wikimedia Foundation. The Wikimedia Foundation is a 501(c)(3) tax-exempt , non-profit, charitable organization registered in Florida, USA . Privacy PolicyAbout WikipediaDisclaimerDeveloperCookie declarationMobile version viewWikimedia Foundation Powered by MediaWiki