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

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