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 formula_1 and formula_2, 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":
where formula_4 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. E.g.,
where formula_6 is the bases in a Cartesian frame. The components of formula_7 can be found by
In a non-Cartesian frame, we do not necessarily have e"i" · for all . However, it is always possible to find a vector e"i" such that
The equality holds when e"i" is the dual base of e"i".
In a Cartesian frame, we have formula_10
Existence and uniqueness.
The dual set always exists and gives an injection from "V" into "V"∗, namely the mapping that sends "vi" 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 for all "i". This map is clearly nonzero on all "vi". If "w" were a finite linear combination of the dual basis vectors "vi", say formula_11 for a finite subset "K" of "I", then for any "j" not in "K", formula_12, 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 and . If one denotes the evaluation of a covector on a vector as a pairing, the biorthogonality condition becomes:
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 formula_14 be a module defined over the ring formula_15 (that is, formula_14 is an object in the category formula_17). Then we define the dual space of formula_14, denoted formula_19, to be formula_20, the module formed of all formula_15-linear module homomorphisms from formula_14 into formula_15. Note then that we may define a dual to the dual, referred to as the double dual of formula_14, written as formula_25, and defined as formula_26.
To formally construct a basis for the dual space, we shall now restrict our view to the case where formula_27 is a finite-dimensional free (left) formula_15-module, where formula_15 is a ring of unity. Then, we assume that the set formula_30 is a basis for formula_27. From here, we define the Kronecker Delta function formula_32 over the basis formula_30 by formula_34 if formula_35 and formula_36 if formula_37. Then the set formula_38 describes a linearly independent set with each formula_39. Since formula_27 is finite-dimensional, the basis formula_30 is of finite cardinality. Then, the set formula_42 is a basis to formula_43 and formula_43 is a free (right) formula_15-module.
Examples.
For example, the standard basis vectors of R2 (the Cartesian plane) are
and the standard basis vectors of its dual space R2* are
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:
where denotes the transpose and
is the volume of the parallelepiped formed by the basis vectors formula_50 and formula_51
In general the dual basis of a basis in a finite dimensional vector space can be readily computed as follows: given the basis formula_52 and corresponding dual basis formula_53 we can build matrices
Then the defining property of the dual basis states that
Hence the matrix for the dual basis formula_56 can be computed as