# Difference between revisions of "Dual basis"

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Given a basis of ''n'' vectors '''e<sub>i</sub>''' spanning the [[direct space]] ''E<sup>n</sup>'', and a vector | Given a basis of ''n'' vectors '''e<sub>i</sub>''' spanning the [[direct space]] ''E<sup>n</sup>'', and a vector | ||

− | '''x''' = ''x<sup>i</sup>'' '''e<sub>i</sub>''', let us consider the ''n'' quantities defined by the | + | '''x''' = ''x<sup> i</sup>'' '''e<sub>i</sub>''', let us consider the ''n'' quantities defined by the |

scalar products of '''x''' with the basis vectors, '''e<sub>i</sub>''': | scalar products of '''x''' with the basis vectors, '''e<sub>i</sub>''': | ||

− | ''x<sub>i</sub>'' = '''x''' . '''e<sub>i</sub>''' = ''x<sup>j</sup>'' '''e<sub>j</sub>''' . '''e<sub>i</sub>''' = ''x<sup>j</sup> g<sub>ji</sub>'', | + | ''x<sub>i</sub>'' = '''x''' . '''e<sub>i</sub>''' = ''x<sup> j</sup>'' '''e<sub>j</sub>''' . '''e<sub>i</sub>''' = ''x<sup> j</sup> g<sub>ji</sub>'', |

where the ''g<sub>ji</sub>'' 's are the doubly covariant components of the [[metric tensor]]. | where the ''g<sub>ji</sub>'' 's are the doubly covariant components of the [[metric tensor]]. | ||

− | By solving these equations in terms of ''x<sup>j</sup>'', one gets: | + | By solving these equations in terms of ''x<sup> j</sup>'', one gets: |

− | ''x | + | ''x<sup> j</sup>'' = ''x<sub>i</sub> g<sup>ij</sup>'' |

− | where the matrix of the ''g | + | where the matrix of the ''g<sup>ij</sup>'' 's is inverse of that of the ''g<sub>ij</sub>'' 's (''g<sup>ik</sup>g<sub>jk</sub>'' = δ<sup>i</sup><sub>j</sub>). The development of vector '''x''' with respect to basis vectors '''e<sub>i</sub>''' can now also be written: |

− | + | '''x''' = ''x<sup> i</sup>'' '''e<sub>i</sub>''' = ''x<sub>i</sub> g<sup>ij</sup>'' '''e<sub>j</sub>''' | |

− | The set of ''n'' vectors '''e | + | The set of ''n'' vectors '''e<sup>i</sup>''' = ''g<sup>ij</sup>'' '''e<sub>j</sub>''' that span the space ''E<sup>n</sup>'' forms a basis since vector '''x''' can be written: |

− | + | '''x''' = ''x<sub>i</sub>'' '''e<sup>i</sup>''' | |

− | This basis is the ''dual basis'' and the ''n'' quantities ''x | + | This basis is the ''dual basis'' and the ''n'' quantities ''x<sub>i</sub>'' defined above are the |

coordinates of '''x''' with respect to the dual basis. In a similar way one can express the direct basis vectors in terms of the dual basis vectors: | coordinates of '''x''' with respect to the dual basis. In a similar way one can express the direct basis vectors in terms of the dual basis vectors: | ||

− | + | '''e<sub>i</sub>''' = ''g<sub>ij</sub>'' '''e<sup>j</sup>''' | |

The scalar products of the basis vectors of the dual and direct bases are: | The scalar products of the basis vectors of the dual and direct bases are: | ||

− | + | ''g<sup>i</sup><sub>j</sub>'' = '''e<sup>i</sup>''' . '''e<sub>j</sub>''' = ''g<sup>ik</sup>'' '''e<sub>k</sub>''' . '''e<sub>j</sub>''' = ''g<sup>ik</sup>g<sub>jk</sub>'' = δ<sup>i</sup><sub>j</sub>. | |

− | One has therefore, since the matrices ''g | + | One has therefore, since the matrices ''g<sup>ik</sup>'' and ''g<sub>ij</sub>'' are inverse: |

− | + | ''g<sup>i</sup><sub>j</sub>'' = '''e<sup>i</sup>''' . '''e<sub>j</sub>''' = δ<sup>i</sup><sub>j</sub>. | |

− | These relations show that the dual basis vectors satisfy the definition conditions of the reciprocal vectors. In a three-dimensional space the dual basis and the basis of [ | + | These relations show that the dual basis vectors satisfy the definition conditions of the reciprocal vectors. In a three-dimensional space the dual basis and the basis of [[reciprocal space]] are identical. |

== Change of basis == | == Change of basis == | ||

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In a change of basis where the direct basis vectors and coordinates transform like: | In a change of basis where the direct basis vectors and coordinates transform like: | ||

− | + | '''e'<sub>j</sub>''' = ''A<sub>j</sub><sup>i</sup>'' '''e<sub>i</sub>'''; ''x'<sup>j</sup>'' = ''B<sub>i</sub><sup> j</sup>'' ''x<sup>i</sup>'', | |

− | where ''A<sub>j</sub><sup>i</sup>'' and ''B<sub>i</sub><sup>j</sup>'' are transformation matrices, transpose of one another, | + | where ''A<sub>j</sub><sup>i</sup>'' and ''B<sub>i</sub><sup> j</sup>'' are transformation matrices, transpose of one another, |

the dual basis vectors '''e<sup>i</sup>''' and the coordinates ''x<sub>i</sub>'' transform according to: | the dual basis vectors '''e<sup>i</sup>''' and the coordinates ''x<sub>i</sub>'' transform according to: | ||

− | + | '''e'<sup>j</sup> ''' = ''B<sub>i</sub><sup> j</sup>'' '''e<sup>i</sup>'''; ''x'<sub>j</sub>'' = ''A<sub>j</sub><sup>i</sup>x<sub>i</sub>''. | |

The coordinates of a vector in reciprocal space are therefore ''covariant'' and the dual basis vectors (or reciprocal vectors) ''contravariant''. | The coordinates of a vector in reciprocal space are therefore ''covariant'' and the dual basis vectors (or reciprocal vectors) ''contravariant''. |

## Revision as of 13:08, 25 January 2006

# Dual basis

### Other languages

Base duale (*Fr*).

## Definition

The dual basis is a basis associated to the basis of a vector space. In three-dimensional space, it is isomorphous to the basis of the reciprocal lattice. It is mathematically defined as follows:

Given a basis of *n* vectors **e _{i}** spanning the direct space

*E*, and a vector

^{n}**x**=

*x*

^{ i}**e**, let us consider the

_{i}*n*quantities defined by the scalar products of

**x**with the basis vectors,

**e**:

_{i}*x _{i}* =

**x**.

**e**=

_{i}*x*

^{ j}**e**.

_{j}**e**=

_{i}*x*,

^{ j}g_{ji}where the *g _{ji}* 's are the doubly covariant components of the metric tensor.

By solving these equations in terms of *x ^{ j}*, one gets:

*x ^{ j}* =

*x*

_{i}g^{ij}where the matrix of the *g ^{ij}* 's is inverse of that of the

*g*'s (

_{ij}*g*= δ

^{ik}g_{jk}^{i}

_{j}). The development of vector

**x**with respect to basis vectors

**e**can now also be written:

_{i}**x** = *x ^{ i}*

**e**=

_{i}*x*

_{i}g^{ij}**e**

_{j}The set of *n* vectors **e ^{i}** =

*g*

^{ij}**e**that span the space

_{j}*E*forms a basis since vector

^{n}**x**can be written:

**x** = *x _{i}*

**e**

^{i}This basis is the *dual basis* and the *n* quantities *x _{i}* defined above are the
coordinates of

**x**with respect to the dual basis. In a similar way one can express the direct basis vectors in terms of the dual basis vectors:

**e _{i}** =

*g*

_{ij}**e**

^{j}The scalar products of the basis vectors of the dual and direct bases are:

*g ^{i}_{j}* =

**e**.

^{i}**e**=

_{j}*g*

^{ik}**e**.

_{k}**e**=

_{j}*g*= δ

^{ik}g_{jk}^{i}

_{j}.

One has therefore, since the matrices *g ^{ik}* and

*g*are inverse:

_{ij}*g ^{i}_{j}* =

**e**.

^{i}**e**= δ

_{j}^{i}

_{j}.

These relations show that the dual basis vectors satisfy the definition conditions of the reciprocal vectors. In a three-dimensional space the dual basis and the basis of reciprocal space are identical.

## Change of basis

In a change of basis where the direct basis vectors and coordinates transform like:

**e' _{j}** =

*A*

_{j}^{i}**e**;

_{i}*x'*=

^{j}*B*

_{i}^{ j}*x*,

^{i}where *A _{j}^{i}* and

*B*are transformation matrices, transpose of one another, the dual basis vectors

_{i}^{ j}**e**and the coordinates

^{i}*x*transform according to:

_{i}**e' ^{j} ** =

*B*

_{i}^{ j}**e**;

^{i}*x'*=

_{j}*A*.

_{j}^{i}x_{i}The coordinates of a vector in reciprocal space are therefore *covariant* and the dual basis vectors (or reciprocal vectors) *contravariant*.

### See also

metric tensor

reciprocal space

The Reciprocal Lattice (Teaching Pamphlet of the *International Union of Crystallography*)

Section 1.1.2 of *International Tables of Crystallography, Volume D*