Espai vectorial quocient

In LINEAR ALGEBRA , the ((((( quotient ))))) of a VECTOR SPACE (( V )) by a SUBSPACE (( N )) is a vector space obtained by "collapsing" (( N )) to zero. The space obtained is called a ((((( quotient space ))))) and is denoted (( V )) /(( N )) (read (( V )) mod (( N )) ). ..

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(==) Definition (==) .. Formally , the construction is as follows Plantilla:Harv .3. Halmos .3. 1974 .3. loc=§21-22. Let (( V )) be a VECTOR SPACE over a FIELD (( K )) , and let (( N )) be a SUBSPACE of (( V )) . We define an EQUIVALENCE RELATION ~ on (( V )) by stating that (( x )) ~ (( y )) if (( x )) &nbsp ;&minus ;&nbsp ;(( y )) &isin ; (( N )) . That is , (( x )) is related to (( y )) if one can be obtained from the other by adding an element of (( N )) . From this definition , one can deduce that any element of (( N )) is equivalent to the zero vector ; in other words all the vectors in (( N )) get mapped into the equivalence class of the zero vector. ..

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The EQUIVALENCE CLASS of (( x )) is often denoted ..

[(( x )) ] = (( x )) + (( N )) ..

since it is given by ..

[(( x )) ] = {(( x )) + (( n ))  : (( n )) &isin ; (( N )) }. ..

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The quotient space (( V )) /(( N )) is then defined as (( V )) /~ , the set of all equivalence classes over (( V )) by ~. Scalar multiplication and addition are defined on the equivalence classes by ..

• &alpha ;[(( x )) ] = [&alpha ;(( x )) ] for all &alpha ; &isin ; (( K )) , and ..
• [(( x )) ]&nbsp ;+&nbsp ;[(( y )) ] = [(( x )) +(( y )) ]. ..

It is not hard to check that these operations are WELL-DEFINED (i.e. do not depend on the choice of representative). These operations turn the quotient space (( V )) /(( N )) into a vector space over (( K )) with (( N )) being the zero class , [0]. ..

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The mapping that associates to (( v )) &nbsp ;&isin ;&nbsp ;(( V )) the equivalence class [(( v )) ] is known as the ((((( quotient map ))))). ..

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(==) Examples (==) .. Let (( X )) &nbsp ;=&nbsp ;((((( R )))))² be the standard Cartesian plane , and let (( Y )) be a line through the origin in (( X )) . Then the quotient space (( X )) /(( Y )) can be identified with the space of all lines in (( X )) which are parallel to (( Y )) . That is to say that , the elements of the set (( X )) /(( Y )) are lines in (( X )) parallel to (( Y )) . This gives one way in which to visualize quotient spaces geometrically. ..

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Another example is the quotient of ((((( R )))))^{(( n ))} by the subspace spanned by the first (( m )) standard basis vectors. The space ((((( R )))))^{(( n ))} consists of all (( n )) -tuples of real numbers ((( x )) ₁ ,… ,(( x )) _{(( n ))} ). The subspace , identified with ((((( R )))))^{(( m ))} , consists of all (( n )) -tuples such that only the first (( m )) entries are non-zero: ((( x )) ₁ ,… ,(( x )) _{(( m ))} ,0 ,0 ,… ,0). Two vectors of ((((( R )))))^{(( n ))} are in the same congruence class modulo the subspace if and only if they are identical in the last (( n )) &minus ;(( m )) coordinates. The quotient space ((((( R )))))^{(( n ))} / ((((( R )))))^{(( m ))} is ISOMORPHIC to ((((( R )))))^{(( n )) &minus ;(( m ))} in an obvious manner. ..

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More generally , if (( V )) is an (internal) DIRECT SUM of subspaces (( U )) and (( W )) : ..

$V=U\oplus W$ ..

then the quotient space (( V )) /(( U )) is naturally isomorphic to (( W )) Plantilla:Harv .3. Halmos .3. 1974 .3. loc=Theorem 22.1. ..

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(==) Properties (==) ..

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There is a natural EPIMORPHISM from (( V )) to the quotient space (( V )) /(( U )) given by sending (( x )) to its equivalence class [(( x )) ]. The KERNEL (or NULLSPACE) of this epimorphism is the subspace (( U )) . This relationship is neatly summarized by the SHORT EXACT SEQUENCE ..

$0\to U\to V\to V/U\to 0.\ ,$ ..

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If (( U )) is a subspace of (( V )) , the DIMENSION of (( V )) /(( U )) is called the ((((( CODIMENSION ))))) of (( U )) in (( V )) . Since a basis of (( V )) may be constructed from a basis (( A )) of (( U )) and a basis (( B )) of (( V )) /(( U )) by adding a representative of each element of (( B )) to (( A )) , the dimension of (( V )) is the sum of the dimensions of (( U )) and (( V )) /(( U )) . If (( V )) is FINITE-DIMENSIONAL , it follows that the codimension of (( U )) in (( V )) is the difference between the dimensions of (( V )) and (( U )) Plantilla:Harv .3. Halmos .3. 1974 .3. loc=Theorem 22.2: ..

$\mathrm {codim} (U)=\dim(V/U)=\dim(V)-\dim(U).$ ..

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Let (( T ))  : (( V )) &rarr ; (( W )) be a LINEAR OPERATOR. The kernel of (( T )) , denoted ker(((( T )))) , is the set of all (( x )) &isin ; (( V )) such that (( Tx )) = 0. The kernel is a subspace of (( V )) . The FIRST ISOMORPHISM THEOREM of linear algebra says that the quotient space (( V )) /ker(((( T )))) is isomorphic to the image of (( V )) in (( W )) . An immediate corollary , for finite-dimensional spaces , is the RANK-NULLITY THEOREM: the dimension of (( V )) is equal to the dimension of the kernel (the (( nullity )) of (( T )) ) plus the dimension of the image (the (( rank )) of (( T )) ). ..

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The COKERNEL of a linear operator (( T ))  : (( V )) &rarr ; (( W )) is defined to be the quotient space (( W )) /im(((( T )))) . ..

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(==) Quotient of a Banach space by a subspace (==) .. If (( X )) is a BANACH SPACE and (( M )) is a CLOSED subspace of (( X )) , then the quotient (( X )) /(( M )) is again a Banach space. The quotient space is already endowed with a vector space structure by the construction of the previous section. We define a norm on (( X )) /(( M )) by ..

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The quotient space (( X )) /(( M )) is COMPLETE with respect to the norm , so it is a Banach space. ..

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(===) Examples (===) .. Let (( C )) [0 ,1] denote the Banach space of continuous real-valued functions on the interval [0 ,1] with the SUP NORM. Denote the subspace of all functions (( f )) &isin ; (( C )) [0 ,1] with (( f )) (0) = 0 by (( M )) . Then the equivalence class of some function (( g )) is determined by its value at 0 , and the quotient space (( C )) [0 ,1]&nbsp ;/&nbsp ;(( M )) is isomorphic to ((((( R ))))). ..

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If (( X )) is a HILBERT SPACE , then the quotient space (( X )) /(( M )) is isomorphic to the HILBERT SPACE#ORTHOGONAL COMPLEMENTS AND PROJECTIONS .3. ORTHOGONAL COMPLEMENT .3. orthogonal complement]] of (( M )) . ..

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(===) Generalization to locally convex spaces (===) .. The quotient of a LOCALLY CONVEX SPACE by a closed subspace is again locally convex Plantilla:Harv .3. Dieudonné .3. 1970 .3. loc=12.14.8. Indeed , suppose that (( X )) is locally convex so that the topology on (( X )) is generated by a family of SEMINORMS {(( p )) _&alpha ; .3. &alpha ;&isin ;(( A )) } where (( A )) is an index set. Let (( M )) be a closed subspace , and define seminorms (( q )) _&alpha by on (( X )) /(( M )) ..

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$q_{\alpha }([x])=\inf _{x\in [x]}p_{\alpha }(x).$ ..

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Then (( X )) /(( M )) is a locally convex space , and the topology on it is the QUOTIENT TOPOLOGY. ..

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If , furthermore , (( X )) is METRIZABLE , then so is (( X )) /(( M )) . If (( X )) is a FRÉCHET SPACE , then so is (( X )) /(( M )) Plantilla:Harv .3. Dieudonné .3. 1970 .3. loc=12.11.3. ..

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(==) See also (==) ..

• QUOTIENT SET ..
• QUOTIENT GROUP ..
• QUOTIENT MODULE ..
• QUOTIENT SPACE (in TOPOLOGY) ..

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(==) References (==) ..

• Plantilla:Citation .3. first=Paul .3. last=Halmos .3. authorlink=Paul Halmos .3. title=Finite dimensional vector spaces .3. publisher=Springer .3. year=1974 .3. isbn=978-0387900933. ..
• Plantilla:Citation .3. first=Jean .3. last=Dieudonné .3. authorlink=Jean Dieudonné .3. title=Treatise on analysis , Volume Ii .3. publisher=Academic Press .3. year=1970. ..

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paraulesenllacos ..

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LINEAR ALGEBRA ..

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VECTOR SPACE ..

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SUBSPACE ..

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VECTOR SPACE ..

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FIELD ..

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SUBSPACE ..

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EQUIVALENCE RELATION ..

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EQUIVALENCE CLASS ..

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WELL-DEFINED ..

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ISOMORPHIC ..

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DIRECT SUM ..

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EPIMORPHISM ..

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KERNEL ..

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NULLSPACE ..

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SHORT EXACT SEQUENCE ..

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DIMENSION ..

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CODIMENSION ..

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FINITE-DIMENSIONAL ..

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LINEAR OPERATOR ..

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FIRST ISOMORPHISM THEOREM ..

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RANK-NULLITY THEOREM ..

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COKERNEL ..

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BANACH SPACE ..

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CLOSED ..

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COMPLETE ..

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SUP NORM ..

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HILBERT SPACE ..

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HILBERT SPACE#ORTHOGONAL COMPLEMENTS AND PROJECTIONS .3. ORTHOGONAL COMPLEMENT ..

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LOCALLY CONVEX SPACE ..

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SEMINORMS ..

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QUOTIENT TOPOLOGY ..

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METRIZABLE ..

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FRÉCHET SPACE ..

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QUOTIENT SET ..

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QUOTIENT GROUP ..

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QUOTIENT MODULE ..

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QUOTIENT SPACE ..

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TOPOLOGY ..