Dual Basis - Examples

Examples

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 Euclidean space, for a given basis {e1, e2, e3}, you can find the biorthogonal (dual) basis by these formulas:


\mathbf{e}^1 = \left(\frac{\mathbf{e}_2\times\mathbf{e}_3}{V}\right)^\text{T},\
\mathbf{e}^2 = \left(\frac{\mathbf{e}_3\times\mathbf{e}_1}{V}\right)^\text{T},\
\mathbf{e}^3 = \left(\frac{\mathbf{e}_1\times\mathbf{e}_2}{V}\right)^\text{T}.

where T denotes the transpose and

V \,=\,
\left(\mathbf{e}_1;\mathbf{e}_2;\mathbf{e}_3\right)\,=\,
\mathbf{e}_1\cdot(\mathbf{e}_2\times\mathbf{e}_3) \,=\,
\mathbf{e}_2\cdot(\mathbf{e}_3\times\mathbf{e}_1) \,=\,
\mathbf{e}_3\cdot(\mathbf{e}_1\times\mathbf{e}_2)

is the volume of the parallelepiped formed by the basis vectors and

Read more about this topic:  Dual Basis

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