TITLE: Tensor Product with Trivial Vector Space
QUESTION [7 upvotes]: This question seems obvious and yet I can't seem to find a good answer anywhere.
Let $V$ be a finite dimensional vector space, and let $0$ denote the trivial vector space. Is $V \otimes 0 = 0$ or $V \otimes 0 = V$? My gut tells me that it is the second case, but in thinking about dimension, tensor product should multiply dimension in which case I think it is the first case.

REPLY [2 votes]: The elements of $V\otimes 0$ are sums of pure tensors of the form ${\bf v}⊗{\bf 0}$. Then, as we can slide scalars across the $\otimes$ symbol, we have ${\bf v}\otimes{\bf 0}={\bf v}\otimes (0\cdot{\bf 0})=(0\cdot{\bf v})\otimes{\bf 0}={\bf 0}\otimes{\bf 0}$.
Thus, there is only one element in $V\otimes 0$, the zero tensor.