Question

linear independence of tensor product of vectors

Answer 1

Let V1,…,Vk be complex vector spaces. Given k vectors v1∈V1,…,vk∈Vk, we define that the tensor product v1⊗…⊗vk has rank 1. For any tensor T∈V1⊗…⊗Vk, the rank of T is the minimum r∈N such that T can be written as a sum of r rank 1 tensors. In this case, there are vectors v1,1,…,vr,1∈V1,…v1,k,…,vr,k∈Vk such that

T=∑i=1rvi,1⊗…⊗vi,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 v1,j,…,vr,j∈Vj, for each j=1…k, and construct the tensor T=∑ri=1vi,1⊗…⊗vi,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=∑ri=1vi,1⊗…⊗vi,k. Is it right to say the vectors v1,j,…,vr,j∈Vj, for each j=1…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.