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
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.
For example, the standard basis vectors of R2 (the Cartesian plane) are
and the standard basis vectors of its dual space R2* are
In 3-dimensional space, for a given basis e you can find the biorthogonal (dual) basis by this formulas:
![e_1^*=\frac{\left[e_2;e_3\right]}{\left(e_1;e_2;e_3\right)}, e_2^*=\frac{\left[e_3;e_1\right]}{\left(e_1;e_2;e_3\right)}, e_3^*=\frac{\left[e_1;e_2\right]}{\left(e_1;e_2;e_3\right)}](http://upload.wikimedia.org/math/b/1/8/b184df3fd3965f066b42817c00aaafc4.png)
