Dual basis

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, …, 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}

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{.}

In 3-dimensional space, for a given basis {e1, e2, e3}, you can find the biorthogonal (dual) basis by these 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)}


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