> [!definition] > > Let $S \in \Lambda^k(V)$ and $T \in \Lambda^l(V)$ be [[Alternating Tensor|alternating tensors]]. Define the **wedge product** $S \wedge T$ by > $ > S \wedge T = {k + l \choose k} \cdot \text{Alt}(S \otimes T) > $ > [!theorem] > > Let $S \in \Lambda^k(V)$, $T \in \Lambda^l(V)$. and $U \in \Lambda^m(V)$, then > $ > (S \wedge T) \wedge U = S \wedge (T \wedge U) = \frac{(k + l + m)!}{k!l!m!} \cdot \text{Alt}(S \otimes T \otimes U) > $ > *Proof*. > $ > \begin{align*} > (S \wedge T) \wedge U &= \frac{(k + l + m)!}{(k + l)!m!}\text{Alt}((S \wedge T) \otimes U) \\ > &= \frac{(k + l + m)!}{(k + l)!m!} \cdot \frac{(k + l)!}{k!l!}\text{Alt}(S \otimes T \otimes U) > \end{align*} > $