I was looking for the definition of the rank of a tensor. I found 2 different definitions depending on whether we use the component-free approach of tensors:
http://en.wikipedia.org/wiki/Tensor:
Definition of the rank: it is the total number of indices required to uniquely select each component, which is equal to the dimension of the array. Hence a scalar is of a tensor of rank 0, a vector is of a tensor of rank 1, a matrix a tensor of rank 2.
http://en.wikipedia.org/wiki/Tensor_(intrinsic_definition) (component-free treatment of tensors):
Definition of the rank in this context: The term rank of a tensor extends the notion of the rank of a matrix in linear algebra. The rank of a matrix A is the smallest number of such outer products that can be summed to produce it: $A = v w^{\mathrm{T}}$.
More generally, the rank of a matrix $A$ is the smallest number of such outer products that can be summed to produce it: $A = v_1w_1^\mathrm{T} + \cdots + > v_kw_k^\mathrm{T}$
Does this mean that the definition of the rank of a tensor depends on whether we use the component-free approach to tensors?