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What is the dual of a vector space?
What is the dual of a vector space?
The dual V' is the vector space of linear maps from V to F, i.e. Hom(V, F).
What is a linear functional?
What is a linear functional?
A linear map from V to F, a member of V' = Hom(V, F).
Define the dual basis in the context of a vector space.
Define the dual basis in the context of a vector space.
If {e_1,..., e_n} is a basis for V, then the dual basis is {e'_i} such that e'_i(e_j) = d_ij, i.e. e'_i(e_j) = 1 if i = j, 0 otherwise.
If V is finite-dimensional, then V is isomorphic to V', and dim(V) = dim(V').
If V is finite-dimensional, then V is isomorphic to V', and dim(V) = dim(V').
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What is the natural isomorphism from V to V''?
What is the natural isomorphism from V to V''?
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What is the definition of an annihilator?
What is the definition of an annihilator?
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What does the statement E(U) is in U^00 imply in finite-dimensional spaces?
What does the statement E(U) is in U^00 imply in finite-dimensional spaces?
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What is a dual map?
What is a dual map?
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When V and W are finite, the dual mapping D: Hom(V, W) -> Hom(W', V') is a natural isomorphism.
When V and W are finite, the dual mapping D: Hom(V, W) -> Hom(W', V') is a natural isomorphism.
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What is the significance of the transpose in the context of dual maps?
What is the significance of the transpose in the context of dual maps?
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Study Notes
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 ).
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Description
Explore the concepts of dual spaces and linear functionals in vector spaces over fields. This quiz delves into the properties of dual bases and the relationship between a vector space and its dual. Test your knowledge on key definitions and theorems related to vector space duality.
