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. 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

$$

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

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

$$

{mathbf{e}_1, mathbf{e}_2} = left{ begin{pmatrix}

 1 

 0

end{pmatrix}, begin{pmatrix}

 0 

 1

end{pmatrix} right} and the standard basis vectors of its dual space R²* are

$$

{mathbf{e}^1, mathbf{e}^2} = left{ begin{pmatrix}

 1 & 0

end{pmatrix}, begin{pmatrix}

 0 & 1

end{pmatrix} right}text{.}