Dual Basis

In linear algebra, a dual basis is a set of vectors that forms a basis for the dual space of a vector space, and forms a biorthogonal system with the basis for the vector space. For a finite-dimensional vector space V, the dual space V* is isomorphic to V, and for any given set of basis vectors {e₁, …, e_n} of V, there is an associated dual basis {e1, …, en} of V* with the relation

where the superscripts of the dual basis elements are indices. Concretely, we can write vectors in an n-dimensional vector space V as n×1 column matrices and elements of the dual space V* as 1×n row matrices that act as linear functionals by left matrix multiplication.