Axioms of the wedge product and characteristics of fields
by math_explorer, Apr 20, 2016, 3:30 PM
Let 
be a vector space over some field 
. Then we can define a new vector space 
whose elements look like 
for 
. The thing is called the wedge product or exterior product of 
and 
. The vector space is called the exterior algebra or Grassmanian algebra, but that's not important. Wikipedia doesn't call it the wedge algebra. Sad.
In fishy terms,
is a product — in more formal terms, it's something you get by quotienting out stuff from a tensor algebra — so you have these properties for all :
where in the last equation, 
is the element of the field . These relations are something you need to travel two Wikipedia articles away from "wedge product" to figure out, but they're not that hard; they're the properties you expect multiplication to have, even if it's hard to visualize what you really get when multiplying two vectors so freely. If you replace with 
, these properties completely define tensor products; but for the wedge product in addition, we have these two properties:
Alternatingness implies anticommutativity: you can get this by expanding
. Anticommutativity almost implies alternatingness: by setting 
, you get ![\[ v \wedge v = -(v \wedge v) \Longrightarrow 2(v \wedge v) = 0, \]](//latex.artofproblemsolving.com/c/e/8/ce81c1cf19e62348900d3bcb058e3b34dbd87373.png)
which implies 
... unless has characteristic 2.
Today's lessons:
is an amazing field, so it's worth it. Maybe that's material for a future post.)
be a vector space over some field 
. Then we can define a new vector space 
whose elements look like 
for 
. The thing is called the wedge product or exterior product of 
and 
. The vector space is called the exterior algebra or Grassmanian algebra, but that's not important. Wikipedia doesn't call it the wedge algebra. Sad.In fishy terms,
is a product — in more formal terms, it's something you get by quotienting out stuff from a tensor algebra — so you have these properties for all :
where in the last equation, 
is the element of the field . These relations are something you need to travel two Wikipedia articles away from "wedge product" to figure out, but they're not that hard; they're the properties you expect multiplication to have, even if it's hard to visualize what you really get when multiplying two vectors so freely. If you replace with 
, these properties completely define tensor products; but for the wedge product in addition, we have these two properties:- Alternatingness(sp?): for all
- Anticommutativity:
for all
for all Alternatingness implies anticommutativity: you can get this by expanding
. Anticommutativity almost implies alternatingness: by setting 
, you get ![\[ v \wedge v = -(v \wedge v) \Longrightarrow 2(v \wedge v) = 0, \]](http://latex.artofproblemsolving.com/c/e/8/ce81c1cf19e62348900d3bcb058e3b34dbd87373.png)
which implies 
... unless has characteristic 2.Today's lessons:
- Alternatingness is a better axiom, and
- Fields of characteristic 2 are weird.
is an amazing field, so it's worth it. Maybe that's material for a future post.)
and codomain 
.
,
,
;![\[ e_1 \wedge e_2, e_1 \wedge e_3, e_1 \wedge e_4, e_2 \wedge e_3, e_2 \wedge e_4, e_3 \wedge e_4. \]](http://latex.artofproblemsolving.com/5/c/7/5c7fdede96620d084f4dd05191f2a950689b7f35.png)
You can break all wedge products into these things: if you consider, say, the wedge product 
(where 
is sloppy notation for the vector with those coordinates), this turns into![\[
(1, 2, 3, 4) \wedge 5e_1 + (1, 2, 3, 4) \wedge 6e_2 + (1, 2, 3, 4) \wedge 7e_3 + (1, 2, 3, 4) \wedge 8e_4 \]](http://latex.artofproblemsolving.com/f/b/2/fb2a0d5fc660a9803bbc58c4a22f2443413173e0.png)
which turns into![\[ 5 e_1 \wedge e_1 + 10 e_2 \wedge e_1 + 15 e_3 \wedge e_1 + \cdots + 32 e_4 \wedge e_4 \]](http://latex.artofproblemsolving.com/b/c/a/bca3a11c4f7841cea83d7a75a939a8f7ba2e2d61.png)
which, if you collect like things and delete things wedged with themselves, turns into a sum of those 6 
basis elements that I am too lazy to write out now.
;
is an element in 
,
,
matrix you get from stacking the vectors together. (You can generalize the wedge product to things like 
times, although this is not what you get just by doing 
two-way wedges; that doesn't even make sense since you can only wedge identical vector spaces. Some people say this is the "right" way to define determinants. I haven't thought about this hard enough to make a conclusion, but I think it's a pretty reasonable opinion.)
if any two of the 
to show that switching the first two slots negates. Other pairs of slots are similar).
to an orthonormal (dot product, though I doubt that's necessary) basis of unit vectors in the top row, and the representations of 
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