Dual of a Vector Space

Dual of a Vector Space

• The dual space ( V' ) consists of linear maps from a vector space ( V ) over a field ( F ) to ( F ).
• Formally expressed as ( \text{Hom}(V, F) ).

Linear Functional

• A linear functional is a specific type of linear map from ( V ) to ( F ).
• Linear functionals form the dual space ( V' = \text{Hom}(V, F) ).

Dual Basis

• If ( {e_1, \ldots, e_n} ) is a basis for ( V ), the corresponding dual basis is ( {e'_i} ).
• Dual basis property: ( e'i(e_j) = \delta{ij} ) (Kronecker delta) where ( e'_i(e_j) = 1 ) if ( i = j ), otherwise 0.
• The effect of a linear map on the basis completely determines the map.

Properties of Dual Space

• For finite-dimensional spaces, there exists an isomorphism between ( V ) and ( V' ).
• The dimensions satisfy ( \text{dim}(V) = \text{dim}(V') ).

Natural Isomorphism from ( V ) to ( V'' )

• For finite-dimensional ( V ), a natural isomorphism ( E: V \to V'' ) maps vectors to their evaluation maps: ( E(v)(f) = f(v) ) for ( f \in V' ).
• The mapping is linear, injective, surjective, hence bijective.
• Independence from the choice of basis characterizes it as a natural isomorphism.

Annihilator

• The annihilator ( U^0 ) of a subspace ( U ) in ( V ) is the set of all linear functionals ( f ) in ( V' ) such that ( f(U) = 0 ).
• ( U^0 ) is a subspace of ( V' ) and is isomorphic to ( (V/U)' ) using the natural isomorphism ( E: U^0 \to V/U ).
• Dimension relation: ( \text{dim}(U^0) = \text{dim}(V) - \text{dim}(U) ).

Properties of Annihilators

• For subspaces ( U ) and ( W ) in ( V ), ( W^0 ) is contained in ( U^0 ) if ( U \subset W ).
• The annihilator of the sum of subspaces ( (U + W)^0 ) equals ( U^0 \cap W^0 ).
• The intersection property: If ( V ) is finite-dimensional, then ( (U \cap W)^0 = U^0 + W^0 ) and both have equal dimensions.

Annihilators and Natural Isomorphism

• Given finite-dimensional ( U \subset V ), the image ( E(U) ) lies in the double annihilator ( U^{00} ).
• If ( v \in U ), then for all ( f \in U^0 ), ( E(v)(f) = 0 ) holds true.

Dual Map

• The dual map ( T': W' \to V' ) is defined for a linear map ( T: V \to W ) such that ( T'(f) = f \circ T ).
• For any ( v \in V ), ( (T'(f))(v) = f(T(v)) ).

Properties of Dual Maps

• The mapping ( D: \text{Hom}(V, W) \to \text{Hom}(W', V') ) defined by ( D(T) = T' ) serves as a natural isomorphism.
• For bases ( e ) in ( V ) and ( f ) in ( W ), the matrix ( M' ) for the dual map ( T' ) is the transpose of the matrix ( M ) for map ( T ): ( M' = M^T ).