Here is a problem I'm stuck with:
Let $V$ be a vector space (on a field $F$) of finite dimension $n$, $v\in V$ and $\mu : V\times V \mapsto V$ a bilinear application determined by its action on the basis vectors: $$\mu(e_i, e_j) = \sum_{k=1}^n \mu_{ijk}e_k.$$ I'm interested in the set of vectors we can generate from $v$ with $\mu$: $$U_v = \{v, \mu(v,v), \mu(v,\mu(v,v)), \mu(\mu(v,v),v)), \mu(\mu(v,v),\mu(v,v)), \dots\}.$$ I would like to find a characterization of the fact that $U_v$ doesn't contain twice the same item, more precisely: looking at the scalars $\mu_{ijk}$, I would like to be able to decide if $$\mu(v_1, v_2) = \mu(v_3, v_4) \Rightarrow v_1 = v_3 \mbox{ and } v_2 = v_4$$ for all $v_1, v_2, v_3, v_4 \in U_v$.
I've tried to look at the matrix representation of $\mu$ (via tensor product) but I can't seem to get anywhere... I'm thinking it could be related to group/algebra theory but I don't really know where to look (I'm an undergrad in CS).
Any pointers?
Thanks.