Theoreom 3.1.1.1 on pg. 68 of Landsberg's "Tensors: Geometry and Applications" is reproduced below:
The first 65 pages of Landberg's book are here: https://pdfs.semanticscholar.org/839b/ad8fab15d0ee73203e36585783b80c6be184.pdf?_ga=2.63959208.629860606.1595715701-1250778599.1595715701
This proof is very interesting as it is a generalization of row rank equals column rank in the language of tensors. However, I am having trouble following the second half of the proof. Are the $a_i$'s in the dual basis of A the same $a_i$'s in the tensor decomposition $T = \sum_{i=1}^r a_i \otimes b_i \otimes c_i$? I believe they have to be in order for the proof to work. However, the $a_i$'s are explicitly stated to not necessarily to linearly independent, and, thus, cannot necessarily be extended to a basis of A.