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, …, en} of V, there is an associated dual basis {e1, …, en} of V* with the relation

\mathbf{e}^i (\mathbf{e}_j) = \begin{cases} 1, & \text{if } i = j \\ 0, & \text{if } i \ne j\text{.} \end{cases}

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.

Read more about Dual Basis:  Examples

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