system
Given the following post, respond as if you're commenting on Reddit:
human
I was just learning about tensors and came across the fact that all scalars are not tensors. What is an example of a scalar that is not a tensor? (Same with vectors, what is a vector that's not a tensor?)
gpt
A tensor algebra is constructed from a vector space V over a field F. If a tensor space is constructed from a vector space V over a field F, then all vectors of V are by definition tensors of rank 1 and all scalars in F, by definition, tensors of rank 0. In that sense every vector and scalar is a tensor.


One can of course think of more elaborate situations, where several vector spaces or fields are involved. In this context if you only construct the tensor algebra of one of these, it might then be reasonable to say that not all vectors (or scalars) are tensors.