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

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 ${\displaystyle V^{*}}$ on a vector in the original space ${\displaystyle V}$:

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

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] For example,

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

where ${\displaystyle \{\mathbf {i} _{1},\mathbf {i} _{2},\mathbf {i} _{3}\}}$ is the basis in a Cartesian frame. The components of ${\displaystyle \mathbf {x} }$ can be found by

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

However, in a non-Cartesian frame, we do not necessarily have ${\displaystyle \mathbf {e} _{i}\cdot \mathbf {e} _{j}=0}$ for all ${\displaystyle i\neq j}$. However, it is always possible to find vectors ${\displaystyle \mathbf {e} ^{i}}$ in the dual space such that

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

The equality holds when the ${\displaystyle \mathbf {e} ^{i}}$s are the dual basis of ${\displaystyle \mathbf {e} _{i}}$s. Notice the difference in position of the index ${\displaystyle i}$.

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 ${\textstyle w=\sum _{i\in K}\alpha _{i}v^{i}}$ for a finite subset K of I, then for any j not in K, ${\textstyle w(v_{j})=\left(\sum _{i\in K}\alpha _{i}v^{i}\right)\left(v_{j}\right)=0}$, 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 dimension (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.

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 with 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 ${\displaystyle \mathbb {R} ^{2}}$ (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 ${\displaystyle (\mathbb {R} ^{2})^{*}}$ 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 ${\displaystyle \{\mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}\}}$, the biorthogonal (dual) basis ${\displaystyle \{\mathbf {e} ^{1},\mathbf {e} ^{2},\mathbf {e} ^{3}\}}$ can be found 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

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

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

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

See also

• Reciprocal lattice
• Miller index
• Zone axis

Notes

References

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