Dual basis


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 edit

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,


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 edit

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 edit

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 edit

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 edit

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 edit

Notes edit

References edit

  • 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.