Thread: Anticommutativity involving odd part of clifford algebras.

1. Anticommutativity involving odd part of clifford algebras.

I want to show that if w∈Cℓ1(V,Q) anticommutes with all v∈V, then w=0. I've been having a bit of trouble though.

2. Re: Anticommutativity involving odd part of clifford algebras.

Originally Posted by idontknowanything
I want to show that if w∈Cℓ1(V,Q) anticommutes with all v∈V, then w=0. I've been having a bit of trouble though.
Hint: Recall that if $\text{char}(k)\ne2$, then there is a certain symmetric bilinear form $\langle-,-\rangle$ in terms of $Q$ such that $uv+vu=2\langle u,v\rangle$ for all $u,v\in V$. Clearly then the result follows. If you aren't aware of this result, this at least tells you which $v$ to plus in to anti commutativity to get that $w=0$ (namely, $w$)