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.
Best Answer
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.