#REDIRECT Vector space
A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. If one considers geometrical vector (spatial), and the operations one can perform upon these vectors such as addition of vectors, scalar multiplication, with some natural constraints such as closure of these operations, associativity of these and combinations of these operations, and so on, we arrive at a description of a mathematical structure which we call a vector space. The "vectors" need not be geometric vectors in the normal sense, but can be any mathematical object that satisfies the following vector space axioms. Polynomials of degree ≤''n'' with real-valued coefficients form a vector space, for example. It is this abstract quality that makes it useful in many areas of modern mathematics. == Formal definition == A vector space over a field (mathematics) ''F'' (such as the field of real number or of complex number numbers) is a set ''V'' together with two operations: * ''vector addition'': ''V'' × ''V'' → ''V'' denoted v + w, where v, w ∈ ''V'', and * ''scalar multiplication'': ''F'' × ''V'' → ''V'' denoted ''a'' v, where ''a'' ∈ ''F'' and v ∈ ''V''. which satisfy following axioms (for all ''a'', ''b'' ∈ ''F'' and u, v, and w ∈ ''V''): # ''V'' is a commutative group under addition of vectors ## There exists an additive identity element 0 in ''V'', such that for all elements v in ''V'', v + 0 = v. ## For all v in V, there exists an element w in V, such that v + w = 0. ## Vector addition is associative: u + (v + w) = (u + v) + w. ## Vector addition is commutative: v + w = w + v. # Scalar multiplication is associative: ''a''(''b'' v) = (''ab'')v. # 1 v = v, where 1 denotes the multiplicative identity in ''F''. # Scalar multiplication Distributivity over vector addition: ''a''(v + w) = ''a'' v + ''a'' w. # Scalar multiplication distributes over scalar addition: (''a'' + ''b'')v = ''a'' v + ''b'' v. The elements of ''V'' are called ''vectors'' and the elements of ''F'' are called ''scalars''. In most applications the field of scalars is the real or complex numbers. * A vector space over the field of real numbers R is called a real vector space. * A vector space over the field of complex numbers C is called a complex vector space. The concept of a vector space is entirely abstract, like the concepts of a group (mathematics), ring (algebra), and field (mathematics). To determine if a set ''V'' is a vector space, one only has to specify the set ''V'', a field ''F'', and define vector addition and scalar multiplication in ''V''. Then, if ''V'' satisfies the above ten axioms, it is a vector space over the field ''F''. ===Elementary properties=== There are a number of properties that follow easily from the vector space axioms. These include: *The zero vector 0 ∈ ''V'' (defined by axiom 3) is unique. *''a'' 0 = 0 for all ''a'' ∈ ''F''. *0 v = 0 for all v ∈ ''V'' where 0 is the additive identity in ''F''. *''a'' v = 0 if and only if either ''a'' = 0 or v = 0. *The additive inverse of a vector v (defined by axiom 4) is unique. It is usually denoted −v. The notation v − w for v + (−w) is also standard. *(−1)v = −v for all v ∈ ''V''. *(−''a'')v = ''a''(−v) = −(''a''v) for all ''a'' ∈ ''F'' and all v ∈ ''V''. == Examples == See ''Examples of vector spaces'' for a list of standard examples. == Subspaces and bases == Given a vector space V, any nonempty subset W of V which is closed under addition and scalar multiplication is called a Linear subspace of V. It is easy to see that subspaces of V are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set of vectors is called their span; if no vector can be removed without diminishing the span, the set is described as being ''linearly independent''. A linearly independent set whose span is the whole space is called a basis (linear algebra). All bases for a given vector space have the same cardinality. Using Zorn's lemma, it can be proved that every vector space has a basis, and vector spaces over a given field are fixed up to isomorphism by a single cardinal number (called the Hamel dimension of the vector space) representing the size of the basis. For instance, the real vector spaces are just R0, R1, R2, R3, …, R∞, …. As you would expect, the dimension of the real vector space R3 is three. A basis makes it possible to express every vector of the space as a unique combination of the field elements. Vector spaces are usually introduced from this coordinatised viewpoint. Given a translationally invariant and rescaling invariant topology over a vector space (preferably infinite-dimensional), the sum of an infinite sequence of vectors can be defined as the limit (topology), if it exists. See topological vector space. == Linear maps == Given two vector spaces V and W over the same field F, one can define linear transformations or “linear maps” from V to W. These are maps from V to W which are compatible with the relevant structure—i.e., they preserve sums and scalar products. The set of all linear maps from V to W, denoted L(V, W), is also a vector space over F. When bases for both V and W are given, linear maps can be expressed in terms of components as matrix (mathematics). An ''isomorphism'' is a linear map that is one-to-one and onto. If there exists an isomorphism between V and W, we call the two spaces ''isomorphic''; they are then essentially identical. The vector spaces over a fixed field F, together with the linear maps, form a category theory. == Generalizations and additional structures == It is common to study vector spaces with certain additional structures. This is often necessary for recovering ordinary notions from geometry. Some of these additional structures include: * A real or complex vector space with a defined length concept, i.e., a norm (mathematics), is called a normed vector space. * A real or complex vector space with a notion of both length and angle is called an inner product space. * A vector space with a topological space compatible with the operations (i.e. such that addition and scalar multiplication are continuous maps) is called a topological vector space. * A vector space with a bilinear operator (defining a multiplication of vectors) is an algebra over a field. The definition of a vector space makes perfectly good sense if one replaces the field of scalars ''F'' by a general ring (mathematics) ''R''. The resulting structure is called a module (mathematics) over ''R''. In other words, a vector space is nothing but a module over a field. == See also == *linear algebra *vector (spatial) - for vectors in physics Abstract algebra Linear algebra Group theory
For the sake of correctness. The * was used in the vector space axioms both as a map * : F x F -> F and as a map * : F x V -> V. I changed a*b to ab. What about the double used + : F x F -> F and + : V x V -> V? -- Georg Muntingh :I wouldn't worry about it. Usually, each multiplication is written without a symbol, and each addition with "+". The reader has to infer from context what operation it is, and this is possible by looking at the elements operated on and asking where they came from. If this is too much for the reader to handle, they probably won't understand it anyway. (Sorry if I sound dismissive.) Otherwise, we would have 4 or 5 different symbols, (+, _, #, *, ...??) and this is just incredibly cluttered. User:Revolver 21:40, 9 Feb 2004 (UTC) To say the same things in more technical language: we have a case here of operator overloading (see also overloading); which is not necessarily a bad thing when types can be inferred. User:Charles Matthews 19:13, 11 Feb 2004 (UTC) How about using + for vector addition and + for field addition? —User:Daniel Brockman 07:16, Mar 8, 2004 (UTC) The section on 'vectors in physics' really belongs at vector (spatial), rather than here. User:Charles Matthews 09:24, 23 Feb 2004 (UTC) ---- I agree about the 'vectors in physics'... Is it really true what is atated "Note that property 5 (commutativity) actually follows from the other 9"? I am almost certain that we need to modify the 4:th axiom (exist -x: x + -x = 0) so that the invers of a vector is commutative in the following way: "exist -x: x + -x = -x + x = 0" for the statement to hold. So I will now modify the 4:th axiom myself.. I would be glad to see a proof of the statement if it was true in its original version. Dj, 14 Mar 2004 :The extra axiom you have added (-x + x = 0) is not needed, as it follows from the first four axioms. Any introductory book on group theory should have a proof, but here's one anyway. Suppose s is an idempotent (that is, s + s = s). Then s = s + 0 = s + (s + -s) = (s + s) + -s = s + -s = 0. So 0 is the only idempotent. For any x we have (-x + x) + (-x + x) = -x + (x + -x) + x = -x + 0 + x = -x + x, that is, -x + x is an idempotent, so -x + x = 0. --User:Zundark 09:09, 14 Mar 2004 (UTC) ---- Sirs: You have a mathematical typesetting error in statement 4 of your formal definition of a vector space over a field F. In the last part of that sentence, when you type v + w = 0, the 0 should be in BOLD FACE, because it is a vector 0, and not a scaler 0. Regards, Harold :You are correct; I've fixed it now. (You could have fixed it yourself.) --User:Zundark 19:07, 26 Jun 2004 (UTC) ---- How does property 5 follow from the other nine? I've added this as an Wikipedia:Open mathematical questions, since it seems to have gone unanswered for several months. User:Prumpf 18:01, 15 Aug 2004 (UTC) :(''x''+''x'')+(''y''+''y'') = (1+1)(''x''+''y'') = (''x''+''y'')+(''x''+''y''), so ''x''+''y'' = ''y''+''x''. The same proof shows more generally that you can't have a "non-abelian module (mathematics)". --User:Zundark 20:01, 15 Aug 2004 (UTC) == Connection == Using the vector space R∞ and a ultrafilter ''U'' on the natural numbers we can creat a hyperreal field! == discussion at Wikipedia talk:WikiProject Mathematics/related articles == This article is part of a series of closely related articles for which I would like to clarify the interrelations. Please contribute your ideas at Wikipedia talk:WikiProject Mathematics/related articles. --User:MarSch 14:08, 12 Jun 2005 (UTC)
See other meanings of words starting from letter:
Words begining with Vector_space:
Vector_Space
Vector_space
Vector_space
Vector_spaces
Vector_space_basis
Vector_space_dimension
Vector_Space_Model
Vector_space_model
Vector Space
#REDIRECT Vector space
Vector space
A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. If one considers geometrical vector (spatial), and the operations one can perform upon these vectors such as addition of vectors, scalar multiplication, with some natural constraints such as closure of these operations, associativity of these and combinations of these operations, and so on, we arrive at a description of a mathematical structure which we call a vector space. The "vectors" need not be geometric vectors in the normal sense, but can be any mathematical object that satisfies the following vector space axioms. Polynomials of degree ≤''n'' with real-valued coefficients form a vector space, for example. It is this abstract quality that makes it useful in many areas of modern mathematics. == Formal definition == A vector space over a field (mathematics) ''F'' (such as the field of real number or of complex number numbers) is a set ''V'' together with two operations: * ''vector addition'': ''V'' × ''V'' → ''V'' denoted v + w, where v, w ∈ ''V'', and * ''scalar multiplication'': ''F'' × ''V'' → ''V'' denoted ''a'' v, where ''a'' ∈ ''F'' and v ∈ ''V''. which satisfy following axioms (for all ''a'', ''b'' ∈ ''F'' and u, v, and w ∈ ''V''): # ''V'' is a commutative group under addition of vectors ## There exists an additive identity element 0 in ''V'', such that for all elements v in ''V'', v + 0 = v. ## For all v in V, there exists an element w in V, such that v + w = 0. ## Vector addition is associative: u + (v + w) = (u + v) + w. ## Vector addition is commutative: v + w = w + v. # Scalar multiplication is associative: ''a''(''b'' v) = (''ab'')v. # 1 v = v, where 1 denotes the multiplicative identity in ''F''. # Scalar multiplication Distributivity over vector addition: ''a''(v + w) = ''a'' v + ''a'' w. # Scalar multiplication distributes over scalar addition: (''a'' + ''b'')v = ''a'' v + ''b'' v. The elements of ''V'' are called ''vectors'' and the elements of ''F'' are called ''scalars''. In most applications the field of scalars is the real or complex numbers. * A vector space over the field of real numbers R is called a real vector space. * A vector space over the field of complex numbers C is called a complex vector space. The concept of a vector space is entirely abstract, like the concepts of a group (mathematics), ring (algebra), and field (mathematics). To determine if a set ''V'' is a vector space, one only has to specify the set ''V'', a field ''F'', and define vector addition and scalar multiplication in ''V''. Then, if ''V'' satisfies the above ten axioms, it is a vector space over the field ''F''. ===Elementary properties=== There are a number of properties that follow easily from the vector space axioms. These include: *The zero vector 0 ∈ ''V'' (defined by axiom 3) is unique. *''a'' 0 = 0 for all ''a'' ∈ ''F''. *0 v = 0 for all v ∈ ''V'' where 0 is the additive identity in ''F''. *''a'' v = 0 if and only if either ''a'' = 0 or v = 0. *The additive inverse of a vector v (defined by axiom 4) is unique. It is usually denoted −v. The notation v − w for v + (−w) is also standard. *(−1)v = −v for all v ∈ ''V''. *(−''a'')v = ''a''(−v) = −(''a''v) for all ''a'' ∈ ''F'' and all v ∈ ''V''. == Examples == See ''Examples of vector spaces'' for a list of standard examples. == Subspaces and bases == Given a vector space V, any nonempty subset W of V which is closed under addition and scalar multiplication is called a Linear subspace of V. It is easy to see that subspaces of V are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set of vectors is called their span; if no vector can be removed without diminishing the span, the set is described as being ''linearly independent''. A linearly independent set whose span is the whole space is called a basis (linear algebra). All bases for a given vector space have the same cardinality. Using Zorn's lemma, it can be proved that every vector space has a basis, and vector spaces over a given field are fixed up to isomorphism by a single cardinal number (called the Hamel dimension of the vector space) representing the size of the basis. For instance, the real vector spaces are just R0, R1, R2, R3, …, R∞, …. As you would expect, the dimension of the real vector space R3 is three. A basis makes it possible to express every vector of the space as a unique combination of the field elements. Vector spaces are usually introduced from this coordinatised viewpoint. Given a translationally invariant and rescaling invariant topology over a vector space (preferably infinite-dimensional), the sum of an infinite sequence of vectors can be defined as the limit (topology), if it exists. See topological vector space. == Linear maps == Given two vector spaces V and W over the same field F, one can define linear transformations or “linear maps” from V to W. These are maps from V to W which are compatible with the relevant structure—i.e., they preserve sums and scalar products. The set of all linear maps from V to W, denoted L(V, W), is also a vector space over F. When bases for both V and W are given, linear maps can be expressed in terms of components as matrix (mathematics). An ''isomorphism'' is a linear map that is one-to-one and onto. If there exists an isomorphism between V and W, we call the two spaces ''isomorphic''; they are then essentially identical. The vector spaces over a fixed field F, together with the linear maps, form a category theory. == Generalizations and additional structures == It is common to study vector spaces with certain additional structures. This is often necessary for recovering ordinary notions from geometry. Some of these additional structures include: * A real or complex vector space with a defined length concept, i.e., a norm (mathematics), is called a normed vector space. * A real or complex vector space with a notion of both length and angle is called an inner product space. * A vector space with a topological space compatible with the operations (i.e. such that addition and scalar multiplication are continuous maps) is called a topological vector space. * A vector space with a bilinear operator (defining a multiplication of vectors) is an algebra over a field. The definition of a vector space makes perfectly good sense if one replaces the field of scalars ''F'' by a general ring (mathematics) ''R''. The resulting structure is called a module (mathematics) over ''R''. In other words, a vector space is nothing but a module over a field. == See also == *linear algebra *vector (spatial) - for vectors in physics Abstract algebra Linear algebra Group theory
Vector space
For the sake of correctness. The * was used in the vector space axioms both as a map * : F x F -> F and as a map * : F x V -> V. I changed a*b to ab. What about the double used + : F x F -> F and + : V x V -> V? -- Georg Muntingh :I wouldn't worry about it. Usually, each multiplication is written without a symbol, and each addition with "+". The reader has to infer from context what operation it is, and this is possible by looking at the elements operated on and asking where they came from. If this is too much for the reader to handle, they probably won't understand it anyway. (Sorry if I sound dismissive.) Otherwise, we would have 4 or 5 different symbols, (+, _, #, *, ...??) and this is just incredibly cluttered. User:Revolver 21:40, 9 Feb 2004 (UTC) To say the same things in more technical language: we have a case here of operator overloading (see also overloading); which is not necessarily a bad thing when types can be inferred. User:Charles Matthews 19:13, 11 Feb 2004 (UTC) How about using + for vector addition and + for field addition? —User:Daniel Brockman 07:16, Mar 8, 2004 (UTC) The section on 'vectors in physics' really belongs at vector (spatial), rather than here. User:Charles Matthews 09:24, 23 Feb 2004 (UTC) ---- I agree about the 'vectors in physics'... Is it really true what is atated "Note that property 5 (commutativity) actually follows from the other 9"? I am almost certain that we need to modify the 4:th axiom (exist -x: x + -x = 0) so that the invers of a vector is commutative in the following way: "exist -x: x + -x = -x + x = 0" for the statement to hold. So I will now modify the 4:th axiom myself.. I would be glad to see a proof of the statement if it was true in its original version. Dj, 14 Mar 2004 :The extra axiom you have added (-x + x = 0) is not needed, as it follows from the first four axioms. Any introductory book on group theory should have a proof, but here's one anyway. Suppose s is an idempotent (that is, s + s = s). Then s = s + 0 = s + (s + -s) = (s + s) + -s = s + -s = 0. So 0 is the only idempotent. For any x we have (-x + x) + (-x + x) = -x + (x + -x) + x = -x + 0 + x = -x + x, that is, -x + x is an idempotent, so -x + x = 0. --User:Zundark 09:09, 14 Mar 2004 (UTC) ---- Sirs: You have a mathematical typesetting error in statement 4 of your formal definition of a vector space over a field F. In the last part of that sentence, when you type v + w = 0, the 0 should be in BOLD FACE, because it is a vector 0, and not a scaler 0. Regards, Harold :You are correct; I've fixed it now. (You could have fixed it yourself.) --User:Zundark 19:07, 26 Jun 2004 (UTC) ---- How does property 5 follow from the other nine? I've added this as an Wikipedia:Open mathematical questions, since it seems to have gone unanswered for several months. User:Prumpf 18:01, 15 Aug 2004 (UTC) :(''x''+''x'')+(''y''+''y'') = (1+1)(''x''+''y'') = (''x''+''y'')+(''x''+''y''), so ''x''+''y'' = ''y''+''x''. The same proof shows more generally that you can't have a "non-abelian module (mathematics)". --User:Zundark 20:01, 15 Aug 2004 (UTC) == Connection == Using the vector space R∞ and a ultrafilter ''U'' on the natural numbers we can creat a hyperreal field! == discussion at Wikipedia talk:WikiProject Mathematics/related articles == This article is part of a series of closely related articles for which I would like to clarify the interrelations. Please contribute your ideas at Wikipedia talk:WikiProject Mathematics/related articles. --User:MarSch 14:08, 12 Jun 2005 (UTC)
See other meanings of words starting from letter:
V
Words begining with Vector_space:
Vector_Space
Vector_space
Vector_space
Vector_spaces
Vector_space_basis
Vector_space_dimension
Vector_Space_Model
Vector_space_model
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