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- A '''vector space''', also known as a '''linear space''', is an abstract [[mathematics|math ...bb{R}^3</math></font> that are relatively easy to visualize, we can make a vector space out of <font style = "vertical-align: 17%"><math>\mathbb{R}^n</math></font15 KB (2,506 words) - 05:16, 11 May 2011
- 12 bytes (1 word) - 17:44, 15 November 2007
- 120 bytes (19 words) - 12:33, 29 November 2008
- {{r|dimension (vector space)}}492 bytes (60 words) - 15:09, 28 July 2009
- The number of elements in any basis for a vector space.91 bytes (14 words) - 02:26, 11 December 2008
Page text matches
- {{rpl|Vector space}}264 bytes (30 words) - 02:32, 25 September 2013
- ...eason often used as a stand-in whenever one expressions and equations in a vector space. ...ily expressed through linear combinations. For instance, a [[basis]] of a vector space can be defined as a set of vectors in the space with the property that ever911 bytes (137 words) - 22:56, 25 November 2008
- * [[inner product]], a generalisation of the above in an abstract vector space.242 bytes (36 words) - 12:38, 31 May 2009
- {{r|Vector space}}423 bytes (60 words) - 15:14, 28 July 2009
- {{r|Vector space}}359 bytes (48 words) - 15:04, 28 July 2009
- #REDIRECT [[vector space]]26 bytes (3 words) - 08:06, 5 November 2008
- In [[mathematics]], a '''norm''' is a function on a [[vector space]] that generalizes to vector spaces the notion of the distance from a point Let ''X'' be a vector space over some subfield ''F'' of the [[complex number|complex numbers]]. Then a880 bytes (157 words) - 22:28, 20 February 2010
- {{r|dimension (vector space)}}492 bytes (60 words) - 15:09, 28 July 2009
- ...ger picture]]) of a quantum mechanical system is postulated to be a unit [[vector space|vector]] (i.e., a vector of norm 1) in some Hilbert space, and physical qua2 KB (258 words) - 12:33, 4 January 2009
- In [[mathematics]], a '''normed space''' is a [[vector space]] that is endowed with a [[norm (mathematics)|norm]]. A [[completeness|comp ...hbb{R}^n</math>. This is the canonical example of a ''finite dimensional'' vector space; in fact ''all'' finite dimensional real normed spaces of dimension ''n'' a982 bytes (148 words) - 07:17, 3 December 2007
- {{r|Vector space}} {{r|Euclidean vector space}}338 bytes (42 words) - 19:08, 5 October 2009
- A vector space that is endowed with an inner product and the corresponding norm.116 bytes (17 words) - 13:40, 4 January 2009
- In [[linear algebra]], a '''basis''' for a [[vector space]] <math>V</math> is a set of [[vector]]s in <math>V</math> such that every ...respect to a basis. Through this transformation, every finite dimensional vector space can be considered to be essentially "the same as" the space <math>\mathbb{R3 KB (464 words) - 19:45, 1 December 2008
- The number of elements in any basis for a vector space.91 bytes (14 words) - 02:26, 11 December 2008
- Matrices that represent the same endomorphism of a vector space with respect to different bases.132 bytes (18 words) - 17:44, 25 November 2008
- A function on a vector space that generalises the notion of the distance from a point of a Euclidean spa157 bytes (26 words) - 15:01, 4 January 2009
- ...books.google.com/books?id=ybwBqELAJDEC&pg=PA71 |chapter=Lecture 2: Tangent vector space |pages=pp. 71 ''ff'' |isbn=0824703855 |year=2001 |publisher=CRC Press}} A s399 bytes (52 words) - 18:07, 20 March 2011
- ...ctors that, in a linear combination, can represent every vector in a given vector space or free module, and such that no element of the set can be represented as a245 bytes (42 words) - 06:20, 4 September 2009
- A vector space endowed with a norm that is complete.88 bytes (13 words) - 16:25, 14 July 2008
- A discrete subgroup of a real vector space.79 bytes (11 words) - 13:32, 29 November 2008
- A vector space that is endowed with a norm.80 bytes (12 words) - 10:18, 4 September 2009
- A bilinear or sesquilinear form on a vector space generalising the dot product in Euclidean spaces.135 bytes (19 words) - 15:24, 28 November 2008
- In [[geometry]], a '''lattice''' is a discrete subgroup of a real [[vector space]].96 bytes (14 words) - 13:26, 1 February 2009
- ...linear operator''') is a [[Function (mathematics)|function]] between two [[Vector space|vector spaces]] that preserves the operations of vector addition and [[Scal The term ''linear transformation'' is especially used for linear maps from a vector space to itself ([[Endomorphism|endomorphisms]]).1 KB (196 words) - 17:24, 20 December 2007
- Vector space of all tangent vectors at a given point of a differentiable manifold.119 bytes (17 words) - 20:19, 4 September 2009
- ...be labeled as 'vectors'. Thus, a vector is defined as a member of ''any'' vector space. Typical vector spaces include the real line, the Euclidean plane or space, ...ace is that superimposed vectors are again elements of the vector space (a vector space is closed under vector addition).4 KB (632 words) - 10:13, 6 January 2010
- ...ebraic structures such as a [[monoid]], [[group (mathematics)|group]] or [[vector space]] have a distinguished element, such as an [[identity element]], and [[morp ** [[Affine space]] versus [[vector space]];1 KB (168 words) - 12:06, 22 November 2008
- ...It is also a [[normed space]] since an inner product induces a norm on the vector space on which it is defined. A [[completeness|complete]] inner product space is1 KB (204 words) - 14:38, 4 January 2009
- A ''Lie algebra'' is a vector space together with a skew-symmetric bilinear operation (the bracket) that fulfil170 bytes (23 words) - 09:27, 27 November 2011
- The property of a system of elements of a module or vector space, that no non-trivial linear combination is zero.149 bytes (23 words) - 16:48, 6 January 2009
- The set of all finite linear combinations of a module over a ring or a vector space over a field.133 bytes (23 words) - 13:20, 6 January 2009
- {{r|Vector space}}1 KB (146 words) - 16:32, 11 January 2010
- {{r|Vector space}}592 bytes (77 words) - 19:15, 11 January 2010
- ...nt space''' of a [[manifold_(geometry)|differentiable manifold]] M is a [[vector space]] at a point p on the manifold whose elements are the tangent vectors (or v ===Directional derivatives as a vector space===4 KB (676 words) - 00:52, 15 November 2007
- ...the minimal polynomial of a [[square matrix]], an [[endomorphism]] of a [[vector space]] or an [[algebraic number]]. ...y dependent]] since the matrix ring has dimension ''n''<sup>2</sup> as a [[vector space]] over ''F'', and so ''A'' satisfies some polynomial. Hence it makes sense4 KB (613 words) - 02:34, 4 January 2013
- {{r|Vector space}}566 bytes (74 words) - 16:25, 11 January 2010
- ...|groups]], [[ring (mathematics)|rings]], [[field (mathematics)|fields]], [[vector space]]s, and [[algebra over a field|algebras]].250 bytes (31 words) - 07:40, 27 July 2008
- In an [[affine space]] or [[vector space]] of [[dimension (vector space)|dimension]] ''n'' we take ''n''+1 points <math>s_0, s_1, \ldots, s_n</math1 KB (187 words) - 03:54, 1 April 2010
- {{r|Dimension (vector space)}}291 bytes (35 words) - 12:54, 31 May 2009
- ...is a [[division ring]], <math>M</math> is called a [[vector space/Advanced|vector space]]. While every nontrivial vector space has a basis, not every module over an arbitrary ring will have a basis. Th2 KB (371 words) - 00:36, 2 February 2009
- ...further [[algebraic structure]] of [[multiplication]] compatible with the vector space structure.1 KB (155 words) - 15:55, 23 December 2008
- {{r|Vector space}}1 KB (172 words) - 15:25, 15 May 2011
- {{r|Vector space}}203 bytes (25 words) - 16:47, 6 January 2009
- ...duct]] in the Euclidean spaces. Among other things, the inner product on a vector space makes it possible to define the geometric operation of projection onto a [[ Let ''X'' be a vector space over a [[field|sub-field]] ''F'' of the [[complex number|complex numbers]].3 KB (511 words) - 00:25, 20 February 2010
- ...or complex [[vector space]]s (or, more generally, in any ''d''-dimensional vector space ''K''<sup>d</sup> over a field ''K'')1 KB (212 words) - 21:14, 9 September 2020
- {{r|vector space}}117 bytes (13 words) - 23:00, 25 November 2008
- ** A finite-[[dimension (vector space)|dimensional]] [[vector space]] over a [[field (algebra)|field]] is a Northerian module.1 KB (213 words) - 17:17, 7 February 2009
- ...''n'' dimensions as an [[affine space]] built on a [[real number|real]] [[vector space]] '''R'''<sup>''n''</sup> then the translations are the maps of the form ...x this point, we find that these must be [[linear map]]s of the underlying vector space which preserve distance. Hence they may be represented by the [[orthogonal3 KB (392 words) - 14:42, 28 November 2008
- ...ct), but it is only defined and makes sense in general for this particular vector space. Both the dot product and the cross product are widely used in in the stud3 KB (575 words) - 12:41, 14 February 2011
- ...echanics in mind it can be employed more generally when working with any [[vector space]].4 KB (690 words) - 12:51, 26 March 2011
- ...omplex, then the dual space ''X''' of <math>\scriptstyle X</math> is the [[vector space]] over ''F'' of all [[continuity|continuous]] linear functionals <math>\scr4 KB (605 words) - 17:25, 20 November 2008
- ...the study of all of these topics can be systematized into the theory of [[vector space]]s.975 bytes (135 words) - 02:13, 3 September 2010
- ...hematicians call a [[self-adjoint operator]]. It is a linear operator on a vector space ''V'' that is equipped with positive definite [[inner product]]. In physic ...ole vector space ''V''. Doing this, physicists assume implicitly that the vector space ''V'' is of finite dimension.8 KB (1,273 words) - 11:29, 9 July 2009
- Let ''R'' be a Reed-Solomon code of length N = 2<sup>m</sup>-1, [[dimension (vector space)|rank]] ''K'' and minimum weight ''N''-''K''+1. The symbols of ''R'' are e2 KB (272 words) - 15:36, 29 October 2008
- Definition: Let <math>V</math> be a [[vector space]] over a [[field]] <math>F</math>, and let <math>\scriptstyle A:V\to V</mat4 KB (731 words) - 17:16, 11 December 2008
- ...'G'', then to each representation of ''G'' on a finite dimensional complex vector space, there is an associated Artin L-function. When ''K'' and ''k'' are [[algeb ...\scriptstyle G = \mathrm{Gal} (K/k)</math> on a finite dimensional complex vector space ''V''. The Artin L-function associated to <math> \rho </math> is defined b2 KB (315 words) - 15:49, 10 December 2008
- In [[linear algebra]], a '''complement''' to a subspace of a vector space is another subspace which forms an internal direct sum. Two such spaces ar1 KB (195 words) - 15:17, 12 December 2008
- A '''vector space''', also known as a '''linear space''', is an abstract [[mathematics|math ...bb{R}^3</math></font> that are relatively easy to visualize, we can make a vector space out of <font style = "vertical-align: 17%"><math>\mathbb{R}^n</math></font15 KB (2,506 words) - 05:16, 11 May 2011
- {{r|Vector space}}679 bytes (85 words) - 18:06, 11 January 2010
- ...''real''' ''n''-'''dimensional affine space''' is distinguished from the [[vector space]] <font style = "vertical-align: 15%"><math>\mathbb{R}^n</math></font> by ...all arrows in a plane can be mapped onto vectors of a ''2''-dimensional [[vector space]] ''V''<sub>''2''</sub>, called the ''difference space'', the plane is a15 KB (2,366 words) - 09:09, 4 April 2010
- ...morphism]] from the Lie algebra to the endomorphisms (linear maps) of some vector space. * Given a matrix Lie algebra '''g'''⊆End(''V'') of some vector space ''V'', then i:'''g'''⊆End(''V'') is a representation. If the dimension o4 KB (670 words) - 21:07, 22 December 2011
- ...red in the theory of [[metric space]]s in topology, the theory of [[normed vector space]]s in functional analysis, and in parts of [[complex analysis]].2 KB (414 words) - 08:12, 16 April 2009
- {{r|vector space}}668 bytes (88 words) - 12:30, 29 November 2008
- {{r|Vector space}}654 bytes (81 words) - 13:36, 29 November 2008
- ...'''module''' is a mathematical structure of which [[abelian group]]s and [[vector space]]s are particular types. They have become ubiquitous in abstract algebra a ...urses usually use real or complex number scalars, the most general type of vector space is a module over a [[division ring]]. The fundamental commonality betwee7 KB (1,154 words) - 02:39, 16 May 2009
- ...he field extension [''E'':''F''] is the [[dimension]] of ''E'' as an ''F''-vector space. The extension '''C'''/'''R''' has degree 2. An extension of degree 2 is3 KB (435 words) - 22:38, 22 February 2009
- : the neutral element in a vector space (the ''zero vector''), or,2 KB (326 words) - 18:28, 17 July 2009
- * [[Linearly independent set]]s in a [[vector space]];2 KB (334 words) - 16:29, 7 February 2009
- ...to itself) under function composition: for example, [[endomorphism]]s of a vector space. Here the idempotents are projections, corresponding to direct sum decompo1,007 bytes (146 words) - 16:14, 13 December 2008
- ...sents the [[identity function]] as a [[linear map|linear operator]] on a [[vector space]].1,020 bytes (136 words) - 10:39, 23 April 2009
- ...under [[similarity]], and hence be defined for an [[endomorphism]] of a [[vector space]], independent of choice of [[basis (linear algebra)|basis]].911 bytes (131 words) - 22:35, 17 February 2009
- ...the '''span''' of a set of elements of a [[module (algebra)|module]] or [[vector space]] is the set of all finite [[linear combination]]s of that set: it may equi968 bytes (162 words) - 13:20, 7 February 2009
- {{r|Vector space}}1 KB (169 words) - 08:53, 22 December 2008
- ...mmetry|symmetries]] or [[transformation]]s. In short a Lie algebra is a [[vector space]] together with a skew-symmetric bilinear operation denoted as bracket that ...invariant and moreover '''k'''-linear and skew-symmetric. This endows the vector space with a bracket.12 KB (1,918 words) - 20:29, 22 December 2011
- ...trace''' is a property of a [[matrix]] and of a [[linear operator]] on a [[vector space]]. The trace plays an important role in the [[representation theory]] of [[ ==Definition for a linear operator on a finite-dimensional vector space==12 KB (1,903 words) - 10:57, 2 February 2009
- {{r|Vector space}}2 KB (206 words) - 19:38, 11 January 2010
- ...procedure or [[algorithm]] for constructing a set of mutually orthogonal [[vector space|vectors]] from a given set of [[linear independence|linearly independent]]2 KB (301 words) - 06:39, 21 October 2007
- ...ension]] of <math>A</math> is equal to the dimension of the <math>k</math>-vector space <math>m/m^2</math>.1 KB (191 words) - 00:03, 21 February 2010
- ...ces, orthogonality, lines, coordinates etc. In a more abstract approach ([[vector space]]s) planes are defined as two-dimensional affine subspaces. In an axiomatic ...ch defines the three-dimensional Euclidean space more algebraically, via [[Vector space|linear spaces]] and quadratic forms, namely, as a real [[affine space]] who16 KB (2,609 words) - 03:09, 8 March 2024
- * In a [[vector space]], multiplication by scalars distributes over addition of vectors.2 KB (226 words) - 13:15, 18 November 2022
- * [[Submodule]]s of a [[module (algebra)]] or [[subspace]]s of a [[vector space]]. The submodule generated by a subset ''A'' may also be obtained as the s2 KB (414 words) - 03:00, 14 February 2010
- ...ces, orthogonality, lines, coordinates etc. In a more abstract approach ([[vector space]]s) planes are defined as two-dimensional affine subspaces. In an axiomatic ...ch defines the three-dimensional Euclidean space more algebraically, via [[Vector space|linear spaces]] and quadratic forms, namely, as a real [[affine space]] who16 KB (2,638 words) - 03:10, 8 March 2024
- ...''n''-dimensional Euclidean space is in one-to-one correspondence to the [[vector space]] ℝ<sup>''n''</sup> consisting of ordered ''n''-tuples (columns) of rea9 KB (1,403 words) - 02:22, 14 October 2013
- In a more abstract approach ([[vector space]]s) lines are defined as one-dimensional affine subspaces. ...ch defines the three-dimensional Euclidean space more algebraically, via [[Vector space|linear spaces]] and quadratic forms, namely, as an [[affine space]] whose d10 KB (1,620 words) - 03:09, 8 March 2024
- In a more abstract approach ([[vector space]]s) lines are defined as one-dimensional affine subspaces. ...ch defines the three-dimensional Euclidean space more algebraically, via [[Vector space|linear spaces]] and quadratic forms, namely, as an [[affine space]] whose d10 KB (1,620 words) - 03:09, 8 March 2024
- ...ess|densely]] defined operator on ''H'' with domain ''D(T)''. Consider the vector space5 KB (914 words) - 08:41, 17 October 2009
- vector space spanned by the eigenstates of <math>\scriptstyle \mathbf{j}^2\otimes 1</mat vector space spanned by the eigenstates of <math>\scriptstyle 1\otimes\mathbf{j}^2</math11 KB (1,759 words) - 10:02, 2 August 2008
- ...1)</math>. It is in fact <math>\mathbf{F}_p</math>-linear on ''F'' as a [[vector space]], with kernel the one-dimensional subspace generated by <math>1_F</math>,2 KB (295 words) - 15:43, 7 December 2008
- ====[[Vector space|Linear]] and [[Topological space|topological]] spaces==== ...space. It is homogeneous. In the words of John Baez, "an affine space is a vector space that's forgotten its origin". A straight line in the affine space is, by de28 KB (4,311 words) - 08:36, 14 October 2010
- ...a [[module (algebra)|module]] over a [[ring (mathematics)|ring]] or of a [[vector space]], is one for which the only [[linear combination]] equal to zero is that f2 KB (307 words) - 16:08, 7 February 2009
- ...K'' is the ''space of test functions''. It can be shown that ''K'' is a [[vector space|linear space]].5 KB (786 words) - 21:28, 19 February 2010
- ...a [[complex vector space]]. In fact, it is a [[locally convex topological vector space]], with the [[norm (mathematics)|seminorms]] being the suprema on [[compact9 KB (1,434 words) - 15:35, 7 February 2009
- ...[[partial differential equation]]s, where <var>n</var> is the [[Dimension (vector space)|dimension]] of the [[Complex space|complex ambient space]] ℂ''<sup>n</su6 KB (874 words) - 03:45, 7 October 2013
- *[[Vector space]]3 KB (496 words) - 22:16, 7 February 2010
- ...e states constitute an orthonormal [[basis (mathematics)|basis]] for the [[vector space]] they belong to. The second-order perturbed energy is12 KB (1,839 words) - 10:43, 5 October 2009
- ...o structure-preserving maps (e.g. two [[group|group homomorphisms]], two [[vector space|linear maps]], two [[metric space|isometries]], two [[manifold|diffeomorphi7 KB (1,151 words) - 14:44, 26 December 2013
- ...II" as a [[roman numeral]]. We can geometrically represent a number with [[vector space|unitless vectors]] in a cartesian system or by drawing simple shapes (e.g.,11 KB (1,701 words) - 20:07, 1 July 2021
- ...theory of the ''Isabelle/HOL'' library. The mathematical definition of a [[vector space]] over a field is formalized as a ''locale:'' locale ''vector space'' =<br>21 KB (3,291 words) - 16:07, 3 November 2013
- ...nics]]. The four matrices form an orthogonal basis for the 4-dimensional [[vector space]] of 2x2 Hermitian matrices.6 KB (1,000 words) - 10:33, 3 February 2011
- ...ear operators on a vector space, which are linear transformations from the vector space into itself.15 KB (2,342 words) - 06:26, 30 November 2011
- ...For instance, aspects of a [[function space]] can be illustrated using a [[vector space]] or a [[topological space]] that introduce interpretations of the 'element13 KB (1,874 words) - 16:11, 4 August 2013
- * '''[[Linear algebra]]''', in which the specific properties of [[vector space]]s are studied (including [[matrix (mathematics)|matrices]]);18 KB (2,669 words) - 08:38, 17 April 2024
- 15 KB (2,211 words) - 06:47, 19 October 2013
- ...h>\scriptstyle \mathbb{C}</math> is a 2-dimensional [[real number|real]] [[vector space]] with respect to the usual operations of addition of complex numbers and m18 KB (3,028 words) - 17:12, 25 August 2013
- ...ant concept here is that of [[vector (spatial)|vector]]s, generalized to [[vector space]]s, and studied in [[linear algebra]]. The study of vectors combines three30 KB (4,289 words) - 16:03, 20 January 2023
- ...g. Many [[algebraic structure]]s such as [[field (algebra)|field]]s and [[vector space]]s are based on groups, and group theory provides an important tool for stu ...pe for the concept of an [[abelian category]], which has applications to [[vector space]]s and beyond.19 KB (3,074 words) - 11:11, 13 February 2009
- ...cinctly, <math>\mathbb{C}</math> is a 2-dimensional [[real number|real]] [[vector space]] with respect to the usual operations of addition of complex numbers and m20 KB (3,304 words) - 17:11, 25 August 2013
- ...ry somewhat because the components may only "predict" but not "map" to the vector space. This produces a statistical principal component use where the most salien16 KB (2,339 words) - 19:24, 29 September 2020
- ...1)</math>. It is in fact <math>\mathbf{F}_2</math>-linear on ''F'' as a [[vector space]].10 KB (1,580 words) - 08:52, 4 March 2009
- .../math>. Choosing a basis <math>g_0,g_1,g_2</math> to the three dimensional vector space <math>H^0(O_C(p+q+r))=\{g:C\to\mathbb{P}^1</math> such that <math>g</math>10 KB (1,637 words) - 16:03, 17 December 2008
- ====Vector space model (relevancy, algebraic, partial match, ranking)==== Variants of the vector space model include:<ref name="isbn0321416910"/>42 KB (5,875 words) - 11:11, 29 March 2012
- ...the case of atomic electrons), and more generally, elements of a complex [[vector space]]. This allows for the calculation of probabilities of outcomes of concrete37 KB (5,578 words) - 04:54, 21 March 2024
- ...is an univalent theory. In contrast, axioms of a linear space (called also vector space) leave a freedom: a linear space may be one-dimensional, two-dimensional, t34 KB (5,174 words) - 21:32, 25 October 2013
- ...additive inverse of its own second derivative. Within the 2-dimensional [[vector space]] ''V'' consisting of all solutions of this equation, the sine function is33 KB (5,179 words) - 08:26, 4 June 2010