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dual basis

In linear algebra, given a vector space ${\\displaystyle V}$ with a basis ${\\displaystyle B}$ of vectors indexed by an index set ${\\displaystyle I}$ (the cardinality of ${\\displaystyle I}$ is the dimension of ${\\displaystyle V}$ ), the dual set of ${\\displaystyle B}$ is a set ${\\displaystyle B^{*}}$ of vectors in the dual space ${\\displaystyle V^{*}}$ with the same index set I such that ${\\displaystyle B}$ and ${\\displaystyle B^{*}}$ form a biorthogonal system. The dual set is always linearly independent but does not necessarily span ${\\displaystyle V^{*}}$ . If it does span ${\\displaystyle V^{*}}$ , then ${\\displaystyle B^{*}}$ is called the dual basis or reciprocal basis for the basis ${\\displaystyle B}$ .
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Dual_basis)

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