As in the case of tensor products, the number of variables of the wedge of two maps is the sum of the numbers of their variables:
Definition:
The wedge product of two vector spaces may be identified with the subspace of their tensor product generated by the skew-symmetric tensors. (This definition, though, works only over fields of characteristic zero. In algebraic work one may need an alternate definition, based on a universal property. This means taking an appropriate quotient of the tensor product, instead - of the same dimension. The difference is harmless for real and complex vector spaces.)
The wedge product of a vector space V with itself k times is called its k-th exterior power and is denoted . If dim V=n, then dim is n-choose-k.
Example:
Let be the dual space of V, i.e. space of all linear maps from V to R.
The second exterior power is the space of all
skew-symmetric bilinear maps from VxV to R.
The definition of an anti-symmetric multilinear operator is an operator
m: Vn -> X such that if there is a linear dependence between
its arguments, the result is 0. Note that the addition of anti-symmetric
operators, or multiplying one by a scalar, is still anti-symmetric --
so the anti-symmetric multilinear operators on Vn form a vector space.
The most famous example of an anti-symmetric
operator is the determinant.
The nth wedge space W, for a module V over
a commutative ring R, together with the anti-symmetric linear wedge operator
w: Vn -> W is such that for every n-linear
anti-symmetric operator
m: Vn -> X there exists a unique linear operator
l: W -> X such that m = l o w. The wedge is unique up to
a unique isomorphism.
One way of defining the wedge space constructively is by dividing the
Tensor space by the subspace generated by all the tensors of n-tuples
which are linearily dependent.
The dimension of the kth wedge space for a free module of dimension
n is n! / (k!(n-k)!).
In particular, that means that up to a constant, there is a single
anti-symmetric functional with the arity of the dimension of the space.
Also note that every linear functional is anti-symmetric.
Note that the wedge operator commutes with the * operator.
In other words, we can define a wedge on functionals such that the result
is an anti-symmetric multilinear functional. In general, we can define the
wedge of an n-linear anti-symmetric functional and an m-linear anti-symmetric
functional to be an (n+m)-linear anti-symmetric functional. Since it turns
out that this operation is associative, we can also define the power
of an anti-symmetric linear functional.
When dealing with differentiable manifolds, we define an "n-form to be
a function from the manifold to the n-th wedge of the cotangent bundle. Such
a form will be said to be differentiable if, when applied to n differentiable
vector fields, the result is a differentiable function.Wedge product of spaces, exterior powers
Definition in generality