Wikipedia / Dual basis
Wikipedia / Dual basis
In linear algebra, given a vector space with a basis of vectors indexed by an index set (the cardinality is the dimension), the dual set of vectors in the dual space form a biorthogonal system. The dual set is always linearly independent but does not necessarily span is called the dual basis or reciprocal basis.
, being biorthogonal means that the elements pair to have an inner product:
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,
can be found by
in the dual space such that
.
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 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
. 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.
-module.
Examples
For example, the standard basis (the Cartesian plane) are
are
In 3-dimensional Euclidean space can be found by formulas below:
where denotes the transpose and
is the volume of the parallelepiped
we can build matrices
Then the defining property of the dual basis states that
can be computed as
See also
Notes
References