Dual basis

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In linear algebra, a dual basis is a set of vectors that forms a basis for the dual space of a 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)= \left\{\begin{matrix} 1, & \mbox{if }i = j \\ 0, & \mbox{if } i \ne j \end{matrix}\right.

Concretely, we can write vectors in an n dimensional vector space V as column (n by 1) matrices and elements of the dual space V* as row (1 by n) matrices that act as linear functionals by left matrix multiplication.

For example, the standard basis vectors of R2 (the Cartesian plane) are

\{\mathbf{e}_1,\mathbf{e}_2\} = \left\{\left(\begin{array}{c} 1 \\ 0 \end{array}\right), \left(\begin{array}{c} 0 \\ 1 \end{array}\right)\right\}

and the standard basis vectors of its dual space R2* are

\{\mathbf{e}^1,\mathbf{e}^2\} = \{\left(\begin{array}{cc} 1 & 0 \end{array}\right), \left(\begin{array}{cc} 0 & 1 \end{array}\right)\}.
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