Is the dot product considered to be between a vector and a covector?

Question

A vector times a covector is of course a scalar. Is the dot product considered to be between a vector and a covector or is it considered to be between 2 vectors?

If between 2 vectors then can I assume that

$\mathbf{u} \cdot \mathbf{v} = \begin{bmatrix} a \\ b \\ c \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix}= \begin{bmatrix} a \\ b \\ c \end{bmatrix} \begin{bmatrix} x & y & z \end{bmatrix}= a x + b y + c z$

where the tensor is a (0,2) tensor?

The obvious example of a dot product is gravity times displacement which give the energy released per mass. Displacement is clearly a vector but gravity has units of energy per distance per mass which sounds like a covector to me.

Answer 1

A (0,2) tensor is a linear function that takes as its input two vectors and outputs a scalar. So the dot product is an example of a (0,2) tensor.

But you can also think of a (0,2) tensor as a function that takes one vector as input and outputs a co-vector. So the metric tensor, which is a generalisation of the dot product, can be thought of as a function that transforms a vector into an equivalent co-vector.