Dual basis

In linear algebra, given a vector space V with a basis B of vectors indexed by an index set I (the cardinality of I is the dimensionality of V), the dual set of B is a set B^∗ of vectors in the dual space V^∗ with the same index set I such that B and B^∗ form a biorthogonal system. The dual set is always linearly independent but does not necessarily span V^∗. If it does span V^∗, then B^∗ is called the dual basis or reciprocal basis for the basis B.

Denoting the indexed vector sets as ${\displaystyle B=\{v_{i}\}_{i\in I}}$ and ${\displaystyle B^{*}=\{v^{i}\}_{i\in I}}$, being biorthogonal means that the elements pair to have an inner product equal to 1 if the indexes are equal, and equal to 0 otherwise. Symbolically, evaluating a dual vector in V^∗ on a vector in the original space V:

${\displaystyle v^{i}\cdot v_{j}=\delta _{j}^{i}={\begin{cases}1&{\text{if }}i=j\\0&{\text{if }}i\neq j{\text{,}}\end{cases}}}$

where ${\displaystyle \delta _{j}^{i}}$ is the Kronecker delta symbol.

Introduction

To perform operations with a vector, we must have a straightforward method of calculating its components. In a Cartesian frame the necessary operation is the dot product of the vector and the base vector.[1] E.g.,

${\displaystyle \mathbf {x} =x^{1}\mathbf {i} _{1}+x^{2}\mathbf {i} _{2}+x^{3}\mathbf {i} _{3}}$

where ${\displaystyle \mathbf {i} _{k}}$ is the bases in a Cartesian frame. The components of ${\displaystyle \mathbf {x} }$ can be found by

${\displaystyle x^{k}=\mathbf {x} \cdot \mathbf {i} _{k}.}$

In a non-Cartesian frame, we do not necessarily have e_i · e_j = 0 for all i ≠ j. However, it is always possible to find a vector eⁱ such that

${\displaystyle x^{i}=\mathbf {x} \cdot \mathbf {e} ^{i}\qquad (i=1,2,3).}$

The equality holds when eⁱ is the dual base of e_i.

In a Cartesian frame, we have ${\displaystyle \mathbf {e} ^{k}=\mathbf {e} _{k}=\mathbf {i} _{k}.}$

Existence and uniqueness

The dual set always exists and gives an injection from V into V^∗, namely the mapping that sends v_i to vⁱ. This says, in particular, that the dual space has dimension greater or equal to that of V.

However, the dual set of an infinite-dimensional V does not span its dual space V^∗. For example, consider the map w in V^∗ from V into the underlying scalars F given by w(v_i) = 1 for all i. This map is clearly nonzero on all v_i. If w were a finite linear combination of the dual basis vectors vⁱ, say ${\displaystyle w=\sum _{i\in K}\alpha _{i}v^{i}}$ for a finite subset K of I, then for any j not in K, ${\displaystyle w(v_{j})=\left(\sum _{i\in K}\alpha _{i}v^{i}\right)\left(v_{j}\right)=0}$, contradicting the definition of w. So, this w does not lie in the span of the dual set.

The dual of an infinite-dimensional space has greater dimensionality (this being a greater infinite cardinality) than the original space has, and thus these cannot have a basis with the same indexing set. However, a dual set of vectors exists, which defines a subspace of the dual isomorphic to the original space. Further, for topological vector spaces, a continuous dual space can be defined, in which case a dual basis may exist.

Finite-dimensional vector spaces

In the case of finite-dimensional vector spaces, the dual set is always a dual basis and it is unique. These bases are denoted by B = { e₁, …, e_n } and B^∗ = { e¹, …, eⁿ }. If one denotes the evaluation of a covector on a vector as a pairing, the biorthogonality condition becomes:

${\displaystyle \left\langle e^{i},e_{j}\right\rangle =\delta _{j}^{i}.}$

The association of a dual basis with a basis gives a map from the space of bases of V to the space of bases of V^∗, and this is also an isomorphism. For topological fields such as the real numbers, the space of duals is a topological space, and this gives a homeomorphism between the Stiefel manifolds of bases of these spaces.

A categorical and algebraic construction of the dual space

Another way to introduce the dual space of a vector space (module) is by introducing it in a categorical sense. To do this, let ${\displaystyle A}$ be a module defined over the ring ${\displaystyle R}$ (that is, ${\displaystyle A}$ is an object in the category ${\displaystyle R{\text{-}}\mathbf {Mod} }$). Then we define the dual space of ${\displaystyle A}$, denoted ${\displaystyle A^{\ast }}$, to be ${\displaystyle {\text{Hom}}_{R}(A,R)}$, the module formed of all ${\displaystyle R}$-linear module homomorphisms from ${\displaystyle A}$ into ${\displaystyle R}$. Note then that we may define a dual to the dual, referred to as the double dual of ${\displaystyle A}$, written as ${\displaystyle A^{\ast \ast }}$, and defined as ${\displaystyle {\text{Hom}}_{R}(A^{\ast },R)}$.

To formally construct a basis for the dual space, we shall now restrict our view to the case where ${\displaystyle F}$ is a finite-dimensional free (left) ${\displaystyle R}$-module, where ${\displaystyle R}$ is a ring of unity. Then, we assume that the set ${\displaystyle X}$ is a basis for ${\displaystyle F}$. From here, we define the Kronecker Delta function ${\displaystyle \delta _{xy}}$ over the basis ${\displaystyle X}$ by ${\displaystyle \delta _{xy}=1}$ if ${\displaystyle x=y}$ and ${\displaystyle \delta _{xy}=0}$ if ${\displaystyle x\neq y}$. Then the set ${\displaystyle S=\lbrace f_{x}:F\to R\;|\;f_{x}(y)=\delta _{xy}\rbrace }$ describes a linearly independent set with each ${\displaystyle f_{x}\in {\text{Hom}}_{R}(F,R)}$. Since ${\displaystyle F}$ is finite-dimensional, the basis ${\displaystyle X}$ is of finite cardinality. Then, the set ${\displaystyle S}$ is a basis to ${\displaystyle F^{\ast }}$ and ${\displaystyle F^{\ast }}$ is a free (right) ${\displaystyle R}$-module.

Examples

For example, the standard basis vectors of R² (the Cartesian plane) are

${\displaystyle \left\{\mathbf {e} _{1},\mathbf {e} _{2}\right\}=\left\{{\begin{pmatrix}1\\0\end{pmatrix}},{\begin{pmatrix}0\\1\end{pmatrix}}\right\}}$

and the standard basis vectors of its dual space R²* are

${\displaystyle \left\{\mathbf {e} ^{1},\mathbf {e} ^{2}\right\}=\left\{{\begin{pmatrix}1&0\end{pmatrix}},{\begin{pmatrix}0&1\end{pmatrix}}\right\}{\text{.}}}$

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

${\displaystyle \mathbf {e} ^{1}=\left({\frac {\mathbf {e} _{2}\times \mathbf {e} _{3}}{V}}\right)^{\mathsf {T}},\ \mathbf {e} ^{2}=\left({\frac {\mathbf {e} _{3}\times \mathbf {e} _{1}}{V}}\right)^{\mathsf {T}},\ \mathbf {e} ^{3}=\left({\frac {\mathbf {e} _{1}\times \mathbf {e} _{2}}{V}}\right)^{\mathsf {T}}.}$

where ^T denotes the transpose and

${\displaystyle V\,=\,\left(\mathbf {e} _{1};\mathbf {e} _{2};\mathbf {e} _{3}\right)\,=\,\mathbf {e} _{1}\cdot (\mathbf {e} _{2}\times \mathbf {e} _{3})\,=\,\mathbf {e} _{2}\cdot (\mathbf {e} _{3}\times \mathbf {e} _{1})\,=\,\mathbf {e} _{3}\cdot (\mathbf {e} _{1}\times \mathbf {e} _{2})}$

is the volume of the parallelepiped formed by the basis vectors ${\displaystyle \mathbf {e} _{1},\,\mathbf {e} _{2}}$ and ${\displaystyle \mathbf {e} _{3}.}$

In general the dual basis of a basis in a finite dimensional vector space can be readily computed as follows: given the basis ${\displaystyle f_{1},\ldots ,f_{n}}$ and corresponding dual basis ${\displaystyle f^{1},\ldots ,f^{n}}$ we can build matrices

${\displaystyle {\begin{aligned}F&={\begin{bmatrix}f_{1}&\cdots &f_{n}\end{bmatrix}}\\G&={\begin{bmatrix}f^{1}&\cdots &f^{n}\end{bmatrix}}\end{aligned}}}$

Then the defining property of the dual basis states that

${\displaystyle G^{T}F=I}$

Hence the matrix for the dual basis ${\displaystyle G}$ can be computed as

${\displaystyle G=(F^{-1})^{T}}$

See also

• Reciprocal lattice
• Miller index
• Zone axis

Notes

1. Lebedev, Cloud & Eremeyev 2010, p. 12.

References

• Lebedev, Leonid P.; Cloud, Michael J.; Eremeyev, Victor A. (2010). Tensor Analysis With Applications to Mechanics. World Scientific. ISBN 978-981431312-4.CS1 maint: ref=harv (link)
• "Finding the Dual Basis". Stack Exchange. May 27, 2012.