My question is about the basics of the exterior algebra. Why is $x\wedge x=0$? I still struggle with the definitions so i can't see why this is trivial.
I'm reffering to http://mathworld.wolfram.com/ExteriorAlgebra.html http://mathworld.wolfram.com/WedgeProduct.html
I am thankful for any help!
Because one of the defining features of exterior algebras is that they anticommute. Specifically, we have for any $x, y$ that $$ x \wedge y = -y \wedge x $$ If we happen to have $y = x$, then we get $$ x \wedge x = -x\wedge x $$ which means that $x \wedge x$ is its own negative, and therefore, unless you are working in characteristic $2$, must be $0$.
To circumvent the "unless you're working in char $2$", the property $x \wedge x = 0$ is really part of the definition of the wedge product.