Dual basis
In linear algebra, given a vector space V {\displaystyle V} with a basis B {\displaystyle B} of vectors indexed by an index set I {\displaystyle I} (the cardinality of I {\displaystyle I} is the dimension of V {\displaystyle V} ), the dual set of B {\displaystyle B} is a set B ∗ {\displaystyle B^{*}} of vectors in the dual space V ∗ {\displaystyle V^{*}} with the same index set I {\displaystyle I} such that B {\displaystyle B} and B ∗ {\displaystyle B^{*}} form a biorthogonal system. The dual set is always linearly independent but does not necessarily span V ∗ {\displaystyle V^{*}} .