Wedge product of basis vectors

[Kevin McHugh]

Is there a set of relationships for the wedge product of basis vectors as there are for the dot product and the cross product?

i.e. e₁*e₁ = 1
e₁*e₂ = 0

e₁ x e₂ = e₃

 

[fresh_42]

It's bilinear, associative and ##v \wedge v=0## (if char ##\mathbb{F} \neq 2##). In general it's ##a \wedge b = (-1)^{nm} b \wedge a## for ##a \in \Lambda^n(V)=
\underbrace{V \wedge V \wedge \ldots \wedge V}_{n\ times}## and ##b \in \Lambda^m(V)= \underbrace{V \wedge V \wedge \ldots \wedge V}_{m\ times}##

 

[Kevin McHugh]

Are those components or basis vectors? Can you explain that in English?

 

[fresh_42]

It is meant as for arbitrary vectors. So especially for basis vectors, too, if you like. In coordinates, it would be some equivalence classes of tensor products, but I don't know how to press this equivalence relation into coordinates. It means basically that the wedge-product or better exterior product is a tensor product, but some tensors are considered to be equal, because the relations below have to be met.

If we have a vector space ##V## over a field ##\mathbb{F}## in which ##1+1 \neq 0## holds, then for ## a,b,c \in V## and ##\lambda \in \mathbb{F}## the following is true:

1. ##(a+b) \wedge c = a\wedge c + b \wedge c##
2. ##\lambda (a \wedge b) = (\lambda a) \wedge b = a \wedge (\lambda b)##
3. ##a \wedge (b \wedge c) = (a \wedge b) \wedge c##
4. ##a \wedge a = 0##
5. ##a_1 \wedge \ldots \wedge a_n \wedge b_1 \wedge \ldots \wedge b_m = (-1)^{nm} b_1 \wedge \ldots \wedge b_m \wedge a_1 \wedge \ldots \wedge a_n##

The first two are called linearity, which together with the fifth becomes multi-linearity (linear in all "factors"), the third one is associativity, the fourth is a special case of the fifth together with the fact, that ##1+1 \neq 0##, which is said as the characteristic of ##\mathbb{F}## is not two, and the fifth alone can be called graduated commutativity, i.e. it determines what happens, if we change the order of "factors". I don't know how to put the formulas in other English words as their names are.

Perhaps you want to read the Wikipedia entry on it: https://en.wikipedia.org/wiki/Exterior_algebra

 

[Stephen Tashi]

Are those components or basis vectors?

The first question is whether ##a \wedge b## is something (e.g. a vector or a scalar) in the same vector space that contains ##a## and ##b##.

I'd say no. For ##a## and ##b## in a a vector space ##V##, in order to define ##a \wedge b##, you must define a different mathematical structure than ##V## itself. ( As an analogy, we can use two real numbers x1, x2 to define an interval [x1,x2], but "an interval" is a different thing than a single real number. )

By contrast, the cross product operation (in 3 dimensions) ##a \times b## does produce a result that is also a element of the the same vector space ##V## that contains ##a## and ##b##.