In linear algebra, given a vector space
form a biorthogonal system.
The dual set is always linearly independent but does not necessarily span
Denoting the indexed vector sets as
Symbolically, evaluating a dual vector in
on a vector in the original space
is the Kronecker delta symbol.
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.
can be found by However, in a non-Cartesian frame, we do not necessarily have
in the dual space such that The equality holds when the
s. Notice the difference in position of the index
The dual set always exists and gives an injection from V into V∗, namely the mapping that sends vi to vi.
For example, consider the map w in V∗ from V into the underlying scalars F given by w(vi) = 1 for all i.
If w were a finite linear combination of the dual basis vectors vi, say
, 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.
In the case of finite-dimensional vector spaces, the dual set is always a dual basis and it is unique.
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.
Another way to introduce the dual space of a vector space (module) is by introducing it in a categorical sense.
be a module defined over the ring
To formally construct a basis for the dual space, we shall now restrict our view to the case where
From here, we define the Kronecker Delta function
describes a linearly independent set with each
(the Cartesian plane) are and the standard basis vectors of its dual space
can be found by formulas below: where T denotes the transpose and is the volume of the parallelepiped formed by the basis vectors
we can build matrices Then the defining property of the dual basis states that Hence the matrix for the dual basis