For the basis of vector fields define the dual basis to be the linear functionals such that
2.
(where are the components of in a particular basis and are the components of in the corresponding dual basis ).
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Further, for topological vector spaces, a continuous dual space can be defined, in which case a dual basis may exist.
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Explicitly, given a Lagrangian subspace ( as defined below ), then a choice of basis defines a dual basis for a complement, by.
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Using a dual basis can provide a way to easily communicate between devices that use different bases, rather than having to explicitly convert between bases using the change of bases formula.
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On the dual basis that this was 37 years ago and I am now 83 years old, I believe I can excuse myself for not recalling my score that first day.
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A basis gives such a form ( via the dual basis ), hence when working on with a Euclidean metric and a fixed orthonormal basis, one can work with only subscripts.
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Furthermore, if a dual basis is implemented then conversion from an element in the original basis to the dual basis can be accomplished with multiplication by the multiplicative identity ( usually 1 ).
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Furthermore, if a dual basis is implemented then conversion from an element in the original basis to the dual basis can be accomplished with multiplication by the multiplicative identity ( usually 1 ).
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A "'dual basis "'is not a concrete basis like the polynomial basis or the normal basis; rather it provides a way of using a second basis for computations.
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