In mathematics, the

**wedge product**, also known as

**exterior product**, is an anti-symmetrisation (alternation) of the tensor product. The wedge product is a distributive associative multiplication of skew-symmetric multilinear maps which is anti-commutative for maps with odd number of variables and commutative otherwise. The systematic theory starts from the exterior power construction for a vector space.

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:**

*k*and

*m*are the numbers of variables for each of the two skew-symmetric functions and alternation of a map is defined to be the signed average of the values over all the permutations of its variables:

## Wedge product of spaces, exterior powers

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 *V*x*V* to **R**.

## Definition in generality

The definition of an *anti-symmetric* multilinear operator is an operator
m: V^{n} -> 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 V^{n} form a vector space.

The most famous example of an anti-symmetric operator is the determinant.

The *n*th wedge space *W*, for a module V over
a commutative ring *R*, together with the anti-symmetric linear wedge operator
*w*: V^{n} -> W is such that for every n-linear
anti-symmetric operator
*m*: V^{n} -> 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 *k*th 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.