Linear algebra concept
In
linear algebra, given a
vector space with a
basis of
vectors indexed by an
index set (the
cardinality of is the
dimension of ), the dual set of is a set of vectors in the
dual space with the same index set such that and form a
biorthogonal system. The dual set is always
linearly independent but does not necessarily
span . If it does span , then is called the dual basis or reciprocal basis for the basis .
Denoting the indexed vector sets as and , 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 on a vector in the original space :
where 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. For example,
where is the basis in a Cartesian frame. The components of can be found by
However, in a non-Cartesian frame, we do not necessarily have for all . However, it is always possible to find vectors in the dual space such that
The equality holds when the s are the dual basis of s. Notice the difference in position of the index .
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 w(vi) = 1 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 for a finite subset K of I, then for any j not in K, , 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.
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 be a module defined over the ring (that is, is an object in the category ). Then we define the dual space of , denoted , to be , the module formed of all -linear module homomorphisms from into . Note then that we may define a dual to the dual, referred to as the double dual of , written as , and defined as .
To formally construct a basis for the dual space, we shall now restrict our view to the case where is a finite-dimensional free (left) -module, where is a ring with unity. Then, we assume that the set is a basis for . From here, we define the Kronecker Delta function over the basis by if and if . Then the set describes a linearly independent set with each . Since is finite-dimensional, the basis is of finite cardinality. Then, the set is a basis to and is a free (right) -module.
Examples
For example, the standard basis vectors of (the
Cartesian plane) are
and the standard basis vectors of its dual space are
In 3-dimensional
Euclidean space, for a given basis , the biorthogonal (dual) basis can be found by formulas below:
where T denotes the
transpose and
is the volume of the
parallelepiped formed by the basis vectors and
In general the dual basis of a basis in a finite-dimensional vector space can be readily computed as follows: given the basis and corresponding dual basis we can build matrices
Then the defining property of the dual basis states that
Hence the matrix for the dual basis can be computed as
See also
Notes
References