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 $\displaystyle \text{char}(k)\ne2$, then there is a certain symmetric bilinear form $\displaystyle \langle-,-\rangle$ in terms of $\displaystyle Q$ such that $\displaystyle uv+vu=2\langle u,v\rangle$ for all $\displaystyle u,v\in V$. Clearly then the result follows. If you aren't aware of this result, this at least tells you which $\displaystyle v$ to plus in to anti commutativity to get that $\displaystyle w=0$ (namely, $\displaystyle w$)