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• Vector_space abstract "A vector space is a mathematical structure formed by a collection of elements called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars in this context. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms, listed below. An example of a vector space is that of Euclidean vectors, which may be used to represent physical quantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, vectors representing displacements in the plane or in three-dimensional space also form vector spaces. Vectors in vector spaces do not necessarily have to be arrow-like objects as they appear in the mentioned examples: vectors are best thought of as abstract mathematical objects with particular properties, which in some cases can be visualized as arrows.Vector spaces are the subject of linear algebra and are well understood from this point of view, since vector spaces are characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. A vector space may be endowed with additional structure, such as a norm or inner product. Such spaces arise naturally in mathematical analysis, mainly in the guise of infinite-dimensional function spaces whose vectors are functions. Analytical problems call for the ability to decide whether a sequence of vectors converges to a given vector. This is accomplished by considering vector spaces with additional structure, mostly spaces endowed with a suitable topology, thus allowing the consideration of proximity and continuity issues. These topological vector spaces, in particular Banach spaces and Hilbert spaces, have a richer theory.Historically, the first ideas leading to vector spaces can be traced back as far as 17th century's analytic geometry, matrices, systems of linear equations, and Euclidean vectors. The modern, more abstract treatment, first formulated by Giuseppe Peano in 1888, encompasses more general objects than Euclidean space, but much of the theory can be seen as an extension of classical geometric ideas like lines, planes and their higher-dimensional analogs.Today, vector spaces are applied throughout mathematics, science and engineering. They are the appropriate linear-algebraic notion to deal with systems of linear equations; offer a framework for Fourier expansion, which is employed in image compression routines; or provide an environment that can be used for solution techniques for partial differential equations. Furthermore, vector spaces furnish an abstract, coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques. Vector spaces may be generalized in several ways, leading to more advanced notions in geometry and abstract algebra.".
• Vector_space thumbnail Vector_addition_ans_scaling.png?width=300.
• Vector_space wikiPageExternalLink ?id=TDQJAAAAIAAJ.
• Vector_space wikiPageExternalLink ?id=bKgAAAAAMAAJ&pg=PA1&dq=Die+Lineale+Ausdehnungslehre+ein+neuer+Zweig+der+Mathematik.
• Vector_space wikiPageExternalLink esla.
• Vector_space wikiPageExternalLink 400338.
• Vector_space wikiPageExternalLink docviewer?did=05230001&seq=9.
• Vector_space wikiPageExternalLink oeitem?id=OE_MOBIUS__1_1_0.
• Vector_space wikiPageExternalLink fm3120.pdf.
• Vector_space wikiPageExternalLink lecture-9-independence-basis-and-dimension.
• Vector_space wikiPageExternalLink www.matrixanalysis.com.
• Vector_space wikiPageExternalLink science?_ob=ArticleURL&_udi=B6WG9-45NJHDR-C&_user=1634520&_coverDate=12%2F31%2F1995&_rdoc=2&_fmt=high&_orig=browse&_srch=doc-info(%23toc%236817%231995%23999779996%23308480%23FLP%23display%23Volume)&_cdi=6817&_sort=d&_docanchor=&_ct=9&_acct=C000054038&_version=1&_urlVersion=0&_userid=1634520&md5=fd995fe2dd19abde0c081f1e989af006.
• Vector_space wikiPageExternalLink science?_ob=ArticleURL&_udi=B6WG9-45NJHDR-D&_user=1634520&_coverDate=12%2F31%2F1995&_rdoc=3&_fmt=high&_orig=browse&_srch=doc-info(%23toc%236817%231995%23999779996%23308480%23FLP%23display%23Volume)&_cdi=6817&_sort=d&_docanchor=&_ct=9&_acct=C000054038&_version=1&_urlVersion=0&_userid=1634520&md5=4327258ef37b4c293b560238058e21ad.
• Vector_space wikiPageExternalLink fulltext.pdf.
• Vector_space wikiPageID "32370".
• Vector_space wikiPageRevisionID "605499176".
• Vector_space hasPhotoCollection Vector_space.
• Vector_space id "T/t092180".
• Vector_space id "p/v096520".
• Vector_space last "BSE-3".
• Vector_space title "Tangent plane".
• Vector_space title "Vector space".
• Vector_space subject Category:Abstract_algebra.
• Vector_space subject Category:Concepts_in_physics.
• Vector_space subject Category:Group_theory.
• Vector_space subject Category:Linear_algebra.
• Vector_space subject Category:Mathematical_structures.
• Vector_space subject Category:Vector_spaces.
• Vector_space subject Category:Vectors.
• Vector_space type Abstraction100002137.
• Vector_space type Attribute100024264.
• Vector_space type Cognition100023271.
• Vector_space type Concept105835747.
• Vector_space type Content105809192.
• Vector_space type FundamentalPhysicsConcepts.
• Vector_space type Idea105833840.
• Vector_space type PsychologicalFeature100023100.
• Vector_space type Quantity105855125.
• Vector_space type Space100028651.
• Vector_space type Variable105857459.
• Vector_space type Vector105864577.
• Vector_space type VectorSpaces.
• Vector_space type Vectors.
• Vector_space comment "A vector space is a mathematical structure formed by a collection of elements called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars in this context. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms, listed below.".
• Vector_space label "Espace vectoriel".
• Vector_space label "Espacio vectorial".
• Vector_space label "Espaço vetorial".
• Vector_space label "Przestrzeń liniowa".
• Vector_space label "Spazio vettoriale".
• Vector_space label "Vector space".
• Vector_space label "Vectorruimte".
• Vector_space label "Vektorraum".
• Vector_space label "Векторное пространство".
• Vector_space label "فضاء متجهي".
• Vector_space label "ベクトル空間".
• Vector_space label "向量空间".
• Vector_space sameAs Vektorový_prostor.
• Vector_space sameAs Vektorraum.
• Vector_space sameAs Διανυσματικός_χώρος.
• Vector_space sameAs Espacio_vectorial.
• Vector_space sameAs Bektore_espazio.
• Vector_space sameAs Espace_vectoriel.
• Vector_space sameAs Ruang_vektor.
• Vector_space sameAs Spazio_vettoriale.
• Vector_space sameAs ベクトル空間.
• Vector_space sameAs 벡터공간.
• Vector_space sameAs Vectorruimte.
• Vector_space sameAs Przestrzeń_liniowa.
• Vector_space sameAs Espaço_vetorial.
• Vector_space sameAs m.07yl0.
• Vector_space sameAs Q125977.
• Vector_space sameAs Q125977.
• Vector_space sameAs Vector_space.
• Vector_space wasDerivedFrom Vector_space?oldid=605499176.
• Vector_space depiction Vector_addition_ans_scaling.png.
• Vector_space isPrimaryTopicOf Vector_space.