How can vector-tensor product be defined?

Question

Given that a tensor $T$ can operate over a vector $u$ as

$$ Tu = (a \otimes b)u = (b \cdot u)a $$

Is it possible to define the oposite operation $uT$ ?

Here's what I've got so far... Let $v$ be a vector, then:

\begin{align*} (uT)v \\&=& u(Tv) && \text{(Assuming this is true)}\\ &=& u \cdot [(a \otimes b)v] && \text{Definition of Tensor}\\ &=& u \cdot [(b\cdot v)a] && \text{Definition of Tensor product}\\ &=& [(b \cdot v)u]\cdot a && \text{Comutativity of the scalar} \\ &=& [(u \otimes b)v]\cdot a && \text{Definition of Tensor product} \\ &=& [Uv]\cdot a && \text{Definition of Tensor} \end{align*}

Well, that doesn't seem very helpful, since now I must know this new tensor $U$ and also must know the vector $a$ that composes $T$ with the tensor product...

I guess I could also try visualizing the whole problem by mixing the matrix representation of tensors, but as an exercise of abstraction, I wanted to avoid doing this... Does someone have any suggestion or tip on how can I proceed to define such operation?

Answer 1

I wanted to be able to operate over $u$ before operating over $v$. In order to do that, I believe it is possible to rewrite the equation like this:

$$ u \cdot Tv = T^Tu \cdot v $$