55-459: (Redirected from Annihilators ) [REDACTED] Look up annihilator in Wiktionary, the free dictionary. Annihilator(s) may refer to: Mathematics [ edit ] Annihilator (ring theory) Annihilator (linear algebra) , the annihilator of a subset of a vector subspace Annihilator method , a type of differential operator, used in
110-521: A n ) {\displaystyle (a_{n})} defines a function where the element ( x n ) {\displaystyle (x_{n})} of R ∞ {\displaystyle \mathbb {R} ^{\infty }} is sent to the number which is a finite sum because there are only finitely many nonzero x n {\displaystyle x_{n}} . The dimension of R ∞ {\displaystyle \mathbb {R} ^{\infty }}
165-567: A ∈ F {\displaystyle a\in F} . Elements of the algebraic dual space V ∗ {\displaystyle V^{*}} are sometimes called covectors , one-forms , or linear forms . The pairing of a functional φ {\displaystyle \varphi } in the dual space V ∗ {\displaystyle V^{*}} and an element x {\displaystyle x} of V {\displaystyle V}
220-405: A 1-dimensional vector space over itself) indexed by A {\displaystyle A} , i.e. there are linear isomorphisms On the other hand, F A {\displaystyle F^{A}} is (again by definition), the direct product of infinitely many copies of F {\displaystyle F} indexed by A {\displaystyle A} , and so
275-508: A function f {\displaystyle f} is identified with the vector in V {\displaystyle V} (the sum is finite by the assumption on f {\displaystyle f} , and any v ∈ V {\displaystyle v\in V} may be written uniquely in this way by the definition of the basis). The dual space of V {\displaystyle V} may then be identified with
330-539: A particular family of parallel lines covering the plane. To compute the value of a functional on a given vector, it suffices to determine which of the lines the vector lies on. Informally, this "counts" how many lines the vector crosses. More generally, if V {\displaystyle V} is a vector space of any dimension, then the level sets of a linear functional in V ∗ {\displaystyle V^{*}} are parallel hyperplanes in V {\displaystyle V} , and
385-426: A particular method for solving differential equations Annihilator matrix , in regression analysis Music [ edit ] Annihilator (band) , a Canadian heavy metal band Annihilator (album) , a 2010 album by the aforementioned band Other media [ edit ] Annihilator (Justice League) , an automaton in the fictional series Justice League Unlimited Annihilators (Marvel Comics) ,
440-427: A real number y {\displaystyle y} . Then, seeing this functional as a matrix M {\displaystyle M} , and x {\displaystyle x} as an n × 1 {\displaystyle n\times 1} matrix, and y {\displaystyle y} a 1 × 1 {\displaystyle 1\times 1} matrix (trivially,
495-399: A real number) respectively, if M x = y {\displaystyle Mx=y} then, by dimension reasons, M {\displaystyle M} must be a 1 × n {\displaystyle 1\times n} matrix; that is, M {\displaystyle M} must be a row vector. If V {\displaystyle V} consists of
550-448: A team of superheroes Annihilator , a 2015 science fiction comic by Grant Morrison and Frazer Irving Annihilator (film) , a 1986 television film starring Mark Lindsay Chapman The Annihilators (film) , a 1985 action film by Charles E. Sellier Jr. The Annihilators (novel) , a 1983 novel by Donald Hamilton See also [ edit ] Annihilation (disambiguation) Annihilating element Topics referred to by
605-426: A unique nondegenerate bilinear form ⟨ ⋅ , ⋅ ⟩ Φ {\displaystyle \langle \cdot ,\cdot \rangle _{\Phi }} on V by Thus there is a one-to-one correspondence between isomorphisms of V to a subspace of (resp., all of) V and nondegenerate bilinear forms on V . If the vector space V is over the complex field, then sometimes it
SECTION 10
#1732858382688660-427: A vector subspace Annihilator method , a type of differential operator, used in a particular method for solving differential equations Annihilator matrix , in regression analysis Music [ edit ] Annihilator (band) , a Canadian heavy metal band Annihilator (album) , a 2010 album by the aforementioned band Other media [ edit ] Annihilator (Justice League) , an automaton in
715-627: A vector to a scalar) such that e 1 ( e 1 ) = 1 {\displaystyle \mathbf {e} ^{1}(\mathbf {e} _{1})=1} , e 1 ( e 2 ) = 0 {\displaystyle \mathbf {e} ^{1}(\mathbf {e} _{2})=0} , e 2 ( e 1 ) = 0 {\displaystyle \mathbf {e} ^{2}(\mathbf {e} _{1})=0} , and e 2 ( e 2 ) = 1 {\displaystyle \mathbf {e} ^{2}(\mathbf {e} _{2})=1} . (Note: The superscript here
770-522: Is R n {\displaystyle \mathbb {R} ^{n}} , if E = [ e 1 | ⋯ | e n ] {\displaystyle E=[\mathbf {e} _{1}|\cdots |\mathbf {e} _{n}]} is a matrix whose columns are the basis vectors and E ^ = [ e 1 | ⋯ | e n ] {\displaystyle {\hat {E}}=[\mathbf {e} ^{1}|\cdots |\mathbf {e} ^{n}]}
825-618: Is countably infinite , whereas R N {\displaystyle \mathbb {R} ^{\mathbb {N} }} does not have a countable basis. This observation generalizes to any infinite-dimensional vector space V {\displaystyle V} over any field F {\displaystyle F} : a choice of basis { e α : α ∈ A } {\displaystyle \{\mathbf {e} _{\alpha }:\alpha \in A\}} identifies V {\displaystyle V} with
880-456: Is a linear map , then the transpose (or dual ) f : W → V is defined by for every φ ∈ W ∗ {\displaystyle \varphi \in W^{*}} . The resulting functional f ∗ ( φ ) {\displaystyle f^{*}(\varphi )} in V ∗ {\displaystyle V^{*}}
935-960: Is a natural homomorphism Ψ {\displaystyle \Psi } from V {\displaystyle V} into the double dual V ∗ ∗ = { Φ : V ∗ → F : Φ l i n e a r } {\displaystyle V^{**}=\{\Phi :V^{*}\to F:\Phi \ \mathrm {linear} \}} , defined by ( Ψ ( v ) ) ( φ ) = φ ( v ) {\displaystyle (\Psi (v))(\varphi )=\varphi (v)} for all v ∈ V , φ ∈ V ∗ {\displaystyle v\in V,\varphi \in V^{*}} . In other words, if e v v : V ∗ → F {\displaystyle \mathrm {ev} _{v}:V^{*}\to F}
990-754: Is a basis of V ∗ {\displaystyle V^{*}} . For example, if V {\displaystyle V} is R 2 {\displaystyle \mathbb {R} ^{2}} , let its basis be chosen as { e 1 = ( 1 / 2 , 1 / 2 ) , e 2 = ( 0 , 1 ) } {\displaystyle \{\mathbf {e} _{1}=(1/2,1/2),\mathbf {e} _{2}=(0,1)\}} . The basis vectors are not orthogonal to each other. Then, e 1 {\displaystyle \mathbf {e} ^{1}} and e 2 {\displaystyle \mathbf {e} ^{2}} are one-forms (functions that map
1045-468: Is a matrix whose columns are the dual basis vectors, then where I n {\displaystyle I_{n}} is the identity matrix of order n {\displaystyle n} . The biorthogonality property of these two basis sets allows any point x ∈ V {\displaystyle \mathbf {x} \in V} to be represented as even when the basis vectors are not orthogonal to each other. Strictly speaking,
1100-425: Is always injective ; and it is always an isomorphism if V {\displaystyle V} is finite-dimensional. Indeed, the isomorphism of a finite-dimensional vector space with its double dual is an archetypal example of a natural isomorphism . Infinite-dimensional Hilbert spaces are not isomorphic to their algebraic double duals, but instead to their continuous double duals. If f : V → W
1155-404: Is an isomorphism if and only if W is finite-dimensional. If V = W then the space of linear maps is actually an algebra under composition of maps , and the assignment is then an antihomomorphism of algebras, meaning that ( fg ) = g f . In the language of category theory , taking the dual of vector spaces and the transpose of linear maps is therefore a contravariant functor from
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#17328583826881210-402: Is called the pullback of φ {\displaystyle \varphi } along f {\displaystyle f} . The following identity holds for all φ ∈ W ∗ {\displaystyle \varphi \in W^{*}} and v ∈ V {\displaystyle v\in V} : where the bracket [·,·] on
1265-408: Is different from Wikidata All article disambiguation pages All disambiguation pages annihilator [REDACTED] Look up annihilator in Wiktionary, the free dictionary. Annihilator(s) may refer to: Mathematics [ edit ] Annihilator (ring theory) Annihilator (linear algebra) , the annihilator of a subset of
1320-399: Is different from Wikidata All article disambiguation pages All disambiguation pages Annihilator (linear algebra) In mathematics , any vector space V {\displaystyle V} has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on V , {\displaystyle V,} together with
1375-472: Is more natural to consider sesquilinear forms instead of bilinear forms. In that case, a given sesquilinear form ⟨·,·⟩ determines an isomorphism of V with the complex conjugate of the dual space The conjugate of the dual space V ∗ ¯ {\displaystyle {\overline {V^{*}}}} can be identified with the set of all additive complex-valued functionals f : V → C such that There
1430-410: Is possible to construct a specific basis in V ∗ {\displaystyle V^{*}} , called the dual basis . This dual basis is a set { e 1 , … , e n } {\displaystyle \{\mathbf {e} ^{1},\dots ,\mathbf {e} ^{n}\}} of linear functionals on V {\displaystyle V} , defined by
1485-417: Is referred to as the bi-orthogonality property . Consider { e 1 , … , e n } {\displaystyle \{\mathbf {e} _{1},\dots ,\mathbf {e} _{n}\}} the basis of V. Let { e 1 , … , e n } {\displaystyle \{\mathbf {e} ^{1},\dots ,\mathbf {e} ^{n}\}} be defined as
1540-610: Is sometimes denoted by a bracket: φ ( x ) = [ x , φ ] {\displaystyle \varphi (x)=[x,\varphi ]} or φ ( x ) = ⟨ x , φ ⟩ {\displaystyle \varphi (x)=\langle x,\varphi \rangle } . This pairing defines a nondegenerate bilinear mapping ⟨ ⋅ , ⋅ ⟩ : V × V ∗ → F {\displaystyle \langle \cdot ,\cdot \rangle :V\times V^{*}\to F} called
1595-485: Is the evaluation map defined by φ ↦ φ ( v ) {\displaystyle \varphi \mapsto \varphi (v)} , then Ψ : V → V ∗ ∗ {\displaystyle \Psi :V\to V^{**}} is defined as the map v ↦ e v v {\displaystyle v\mapsto \mathrm {ev} _{v}} . This map Ψ {\displaystyle \Psi }
1650-945: Is the index, not an exponent.) This system of equations can be expressed using matrix notation as Solving for the unknown values in the first matrix shows the dual basis to be { e 1 = ( 2 , 0 ) , e 2 = ( − 1 , 1 ) } {\displaystyle \{\mathbf {e} ^{1}=(2,0),\mathbf {e} ^{2}=(-1,1)\}} . Because e 1 {\displaystyle \mathbf {e} ^{1}} and e 2 {\displaystyle \mathbf {e} ^{2}} are functionals, they can be rewritten as e 1 ( x , y ) = 2 x {\displaystyle \mathbf {e} ^{1}(x,y)=2x} and e 2 ( x , y ) = − x + y {\displaystyle \mathbf {e} ^{2}(x,y)=-x+y} . In general, when V {\displaystyle V}
1705-426: Is the sequence consisting of all zeroes except in the i {\displaystyle i} -th position, which is 1. The dual space of R ∞ {\displaystyle \mathbb {R} ^{\infty }} is (isomorphic to) R N {\displaystyle \mathbb {R} ^{\mathbb {N} }} , the space of all sequences of real numbers: each real sequence (
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1760-490: The (algebraic) dual space V ∗ {\displaystyle V^{*}} (alternatively denoted by V ∨ {\displaystyle V^{\lor }} or V ′ {\displaystyle V'} ) is defined as the set of all linear maps φ : V → F {\displaystyle \varphi :V\to F} ( linear functionals ). Since linear maps are vector space homomorphisms ,
1815-475: The natural pairing . If V {\displaystyle V} is finite-dimensional, then V ∗ {\displaystyle V^{*}} has the same dimension as V {\displaystyle V} . Given a basis { e 1 , … , e n } {\displaystyle \{\mathbf {e} _{1},\dots ,\mathbf {e} _{n}\}} in V {\displaystyle V} , it
1870-424: The above statement only makes sense once the inner product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } and the corresponding duality pairing are introduced, as described below in § Bilinear products and dual spaces . In particular, R n {\displaystyle \mathbb {R} ^{n}} can be interpreted as
1925-682: The action of a linear functional on a vector can be visualized in terms of these hyperplanes. If V {\displaystyle V} is not finite-dimensional but has a basis e α {\displaystyle \mathbf {e} _{\alpha }} indexed by an infinite set A {\displaystyle A} , then the same construction as in the finite-dimensional case yields linearly independent elements e α {\displaystyle \mathbf {e} ^{\alpha }} ( α ∈ A {\displaystyle \alpha \in A} ) of
1980-462: The basis of V {\displaystyle V} , and any function θ : A → F {\displaystyle \theta :A\to F} (with θ ( α ) = θ α {\displaystyle \theta (\alpha )=\theta _{\alpha }} ) defines a linear functional T {\displaystyle T} on V {\displaystyle V} by Again,
2035-406: The bilinear form determines a linear mapping defined by If the bilinear form is nondegenerate , then this is an isomorphism onto a subspace of V . If V is finite-dimensional, then this is an isomorphism onto all of V . Conversely, any isomorphism Φ {\displaystyle \Phi } from V to a subspace of V (resp., all of V if V is finite dimensional) defines
2090-481: The dual is given by the Erdős–Kaplansky theorem . If V is finite-dimensional, then V is isomorphic to V . But there is in general no natural isomorphism between these two spaces. Any bilinear form ⟨·,·⟩ on V gives a mapping of V into its dual space via where the right hand side is defined as the functional on V taking each w ∈ V to ⟨ v , w ⟩ . In other words,
2145-416: The dual space is an important concept in functional analysis . Early terms for dual include polarer Raum [Hahn 1927], espace conjugué , adjoint space [Alaoglu 1940], and transponierter Raum [Schauder 1930] and [Banach 1932]. The term dual is due to Bourbaki 1938. Given any vector space V {\displaystyle V} over a field F {\displaystyle F} ,
2200-583: The dual space may be denoted hom ( V , F ) {\displaystyle \hom(V,F)} . The dual space V ∗ {\displaystyle V^{*}} itself becomes a vector space over F {\displaystyle F} when equipped with an addition and scalar multiplication satisfying: for all φ , ψ ∈ V ∗ {\displaystyle \varphi ,\psi \in V^{*}} , x ∈ V {\displaystyle x\in V} , and
2255-548: The dual space, but they will not form a basis. For instance, consider the space R ∞ {\displaystyle \mathbb {R} ^{\infty }} , whose elements are those sequences of real numbers that contain only finitely many non-zero entries, which has a basis indexed by the natural numbers N {\displaystyle \mathbb {N} } . For i ∈ N {\displaystyle i\in \mathbb {N} } , e i {\displaystyle \mathbf {e} _{i}}
Annihilator - Misplaced Pages Continue
2310-531: The fictional series Justice League Unlimited Annihilators (Marvel Comics) , a team of superheroes Annihilator , a 2015 science fiction comic by Grant Morrison and Frazer Irving Annihilator (film) , a 1986 television film starring Mark Lindsay Chapman The Annihilators (film) , a 1985 action film by Charles E. Sellier Jr. The Annihilators (novel) , a 1983 novel by Donald Hamilton See also [ edit ] Annihilation (disambiguation) Annihilating element Topics referred to by
2365-722: The following: e i ( c 1 e 1 + ⋯ + c n e n ) = c i , i = 1 , … , n {\displaystyle \mathbf {e} ^{i}(c^{1}\mathbf {e} _{1}+\cdots +c^{n}\mathbf {e} _{n})=c^{i},\quad i=1,\ldots ,n} . These are a basis of V ∗ {\displaystyle V^{*}} because: and { e 1 , … , e n } {\displaystyle \{\mathbf {e} ^{1},\dots ,\mathbf {e} ^{n}\}} generates V ∗ {\displaystyle V^{*}} . Hence, it
2420-426: The identification is a special case of a general result relating direct sums (of modules ) to direct products. If a vector space is not finite-dimensional, then its (algebraic) dual space is always of larger dimension (as a cardinal number ) than the original vector space. This is in contrast to the case of the continuous dual space, discussed below, which may be isomorphic to the original vector space even if
2475-889: The latter is infinite-dimensional. The proof of this inequality between dimensions results from the following. If V {\displaystyle V} is an infinite-dimensional F {\displaystyle F} -vector space, the arithmetical properties of cardinal numbers implies that where cardinalities are denoted as absolute values . For proving that d i m ( V ) < d i m ( V ∗ ) , {\displaystyle \mathrm {dim} (V)<\mathrm {dim} (V^{*}),} it suffices to prove that | F | ≤ | d i m ( V ∗ ) | , {\displaystyle |F|\leq |\mathrm {dim} (V^{\ast })|,} which can be done with an argument similar to Cantor's diagonal argument . The exact dimension of
2530-418: The left is the natural pairing of V with its dual space, and that on the right is the natural pairing of W with its dual. This identity characterizes the transpose, and is formally similar to the definition of the adjoint . The assignment f ↦ f produces an injective linear map between the space of linear operators from V to W and the space of linear operators from W to V ; this homomorphism
2585-469: The relation for any choice of coefficients c i ∈ F {\displaystyle c^{i}\in F} . In particular, letting in turn each one of those coefficients be equal to one and the other coefficients zero, gives the system of equations where δ j i {\displaystyle \delta _{j}^{i}} is the Kronecker delta symbol. This property
2640-419: The same term [REDACTED] This disambiguation page lists articles associated with the title Annihilator . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Annihilator&oldid=1247377402 " Category : Disambiguation pages Hidden categories: Short description
2695-419: The same term [REDACTED] This disambiguation page lists articles associated with the title Annihilator . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Annihilator&oldid=1247377402 " Category : Disambiguation pages Hidden categories: Short description
2750-523: The space F A {\displaystyle F^{A}} of all functions from A {\displaystyle A} to F {\displaystyle F} : a linear functional T {\displaystyle T} on V {\displaystyle V} is uniquely determined by the values θ α = T ( e α ) {\displaystyle \theta _{\alpha }=T(\mathbf {e} _{\alpha })} it takes on
2805-455: The space ( F A ) 0 {\displaystyle (F^{A})_{0}} of functions f : A → F {\displaystyle f:A\to F} such that f α = f ( α ) {\displaystyle f_{\alpha }=f(\alpha )} is nonzero for only finitely many α ∈ A {\displaystyle \alpha \in A} , where such
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#17328583826882860-508: The space of columns of n {\displaystyle n} real numbers , its dual space is typically written as the space of rows of n {\displaystyle n} real numbers. Such a row acts on R n {\displaystyle \mathbb {R} ^{n}} as a linear functional by ordinary matrix multiplication . This is because a functional maps every n {\displaystyle n} -vector x {\displaystyle x} into
2915-474: The space of geometrical vectors in the plane, then the level curves of an element of V ∗ {\displaystyle V^{*}} form a family of parallel lines in V {\displaystyle V} , because the range is 1-dimensional, so that every point in the range is a multiple of any one nonzero element. So an element of V ∗ {\displaystyle V^{*}} can be intuitively thought of as
2970-453: The sum is finite because f α {\displaystyle f_{\alpha }} is nonzero for only finitely many α {\displaystyle \alpha } . The set ( F A ) 0 {\displaystyle (F^{A})_{0}} may be identified (essentially by definition) with the direct sum of infinitely many copies of F {\displaystyle F} (viewed as
3025-751: The vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the algebraic dual space . When defined for a topological vector space , there is a subspace of the dual space, corresponding to continuous linear functionals , called the continuous dual space . Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures , distributions , and Hilbert spaces . Consequently,
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