Let $V_1, \ldots, V_k$ be complex vector spaces. Given $k$ vectors $v_1 \in V_1, \ldots, v_k \in V_k$, we define that the tensor product $v_1 \otimes \ldots \otimes v_k$ has rank 1. For any tensor $T \in V_1 \otimes \ldots \otimes V_k$, the rank of $T$ is the minimum $r \in \mathbb{N}$ such that $T$ can be written as a sum of $r$ rank 1 tensors. In this case, there are vectors $v_{1,1}, \ldots, v_{r,1} \in V_1, \ldots v_{1,k}, \ldots, v_{r,k} \in V_k$ such that
$$T = \sum_{i=1}^r v_{i,1} \otimes \ldots \otimes v_{i,k}.$$
I have two questions about the relation between this decomposition and the linear dependency of the vectors.
1) Suppose we have linearly independent vectors $v_{1,j}, \ldots,v_{r,j} \in V_j$, for each $j=1 \ldots k$, and construct the tensor $T = \sum_{i=1}^r v_{i,1} \otimes \ldots \otimes v_{i,k}$. Is it right write to say the rank of $T$ is $r$? If not, what conditions should be considered instead just independence?
2) On the other hand, suppose we know $T$ has rank $r$ and can be written as $T = \sum_{i=1}^r v_{i,1} \otimes \ldots \otimes v_{i,k}$. Is it right to say the vectors $v_{1,j}, \ldots,v_{r,j} \in V_j$, for each $j=1 \ldots k$, are linearly independent?
I'm aware that tensors are not so simple and probably these relations doesn't hold. In this case I'm also accepting suggestions in the following sense:
1) What properties the vectors should have in order to construct a tensor of rank $r$.
2) In the case we already have a tensor of rank $r$ (together with its decomposition), what properties the vectors forming it should have?
Thank you.