We have vectors that have components in the form



that are scalars for basis of the form



Then we have simple q-vectors that have components



that are scalars for basis of the form



Then we have covectors, or 1-forms



with basis



and simple p-forms



with basis




Now we get these tensor things, which we are only given the component form, something like



Am I correct in assuming the basis of such an animal is something like



?

Or would it be something like



?

Or some other permutation of vectors and 1-forms?

No.

Tensors live in a space like this

where is a vector space and is its dual. Note that while Grassmann algebras deal with antisymmetric multilinear forms, tensors are just multilinear and in general will split into a symmetric and antisymmetric part.

Once we get to differentiable manifolds then will be the tangent space at a given point on the manifold and will be the corresponding cotangent space. A typical (p,q) basis element will be