Can the tensor product be visualized as a machine for processing vectors?

[Mandelbroth]

"Seeing" Tensor Products

Is there a way to "visualize" the tensor product of two (or $n$) vectors/tensors/algebras/etc.?

I'm having a lot of trouble making the tensor product feel intuitive. I know its properties, and I can usually apply it without too much of a problem, but it does not feel "easy." Any ideas?

 

[PhysicsandSuch]

For tensor actions in general, I always started by thinking about the properties in lower dimensional spaces, where they can make more physical sense or can be mapped out/drawn explicitly on paper. It tends to bolster your intuition for what happens with higher rank objects.

Then, you can just think of a general action as something that you can already visualize, but in a projected sense (ie, you are imagining only a part of the true behavior, as more exists "behind the scenes," but it is enough to know what's going on).

 

[Chestermiller]

I'm probably coming at this from a different perspective from you, but maybe this can help. My background is in engineering. I like to visualize that tensor product as a machine for processing vectors. You feed a vector to the machine, and the tensor product maps the vector into a new vector. If $\vec{L}\otimes \vec{R}$ is the vector product of the vectors $\vec{L}$ and $\vec{R}$, then if I dot the vector product on the right by a vector $\vec{V}$, I get:

$$(\vec{L}\otimes \vec{R})\centerdot\vec{V}=\vec{L}(\vec{R}\centerdot\vec{V})$$
This is a vector in the direction of $\vec{L}$, with a magnitude equal to the magnitude of $\vec{L}$ times $(\vec{R}\centerdot\vec{V})$. If I dot the vector product on the left by a vector $\vec{V}$, I get an analogous result.

As a physicist or mathematician, I don't know if this makes any sense to you. But to me as an engineer it makes perfect sense, and I have used it many time in practice when working with the engineering stress tensor to map a unit normal vector to a surface into the traction vector.

Chet