Two definitions of vectors

Question

Abstractly, are vectors better defined as elements of a vector space or rank 1 tensors. I've heard of vectors in the context of tensors as generalizations of scalars, vectors, and matrices. However, since you could have a vector space of matrices, matrices can be considered vectors sometimes. These definitions don't seem to compile since one separates the ideas.

Answer 1

Maybe type (1,0) tensor is better than rank 1 tensor for describing vectors. Co-vectors are also rank 1 tensors. By definition, a type (m,n) tensor is an element of $\underbrace{V\otimes\cdots\otimes V}_m\otimes \underbrace{V^\ast\otimes\cdots V^\ast}_n.$ Saying something is a type (1,0) tensor is equivalent to saying it is a member of a vector space. By tradition, a rank 0 tensor just represents a member of the field that the vector space $V$ is over. For clarity sake, it might be "better" to just say that vectors are members are vector spaces and tensors are a type of equivalence class over vector and dual spaces. $V\otimes W=F\!\left(V\times W\right)/\sim$ where $F$ is the free vector space and $\sim$ is the equivalence relation that defines the tensor product. When combined with scaling and addition rules, it has a basis of $\left[v,w\right]$ for $v\in V, w\in W$ usually denoted $v\otimes w.$