So I am looking for a way to "couple" a scalar and a vector such that "events" on the scalar affect the vector.
That sounds weird, let me elaborate.
Let us say we have a tuple $(s, v)$ where $s$ is a scalar and $v$ a vector, $s$ is not just another entry of $v$ it is special, what happens to $s$ must happen to $v$ in a general sense.
For example let us say we have 2 objects on this representation $a = (s_1, v_1), b = (s_2, v_2)$. Let us say we make a linear combination of $s_1, s_2$ as $u s_1 + v s_2$.
Then the same must happen to their entangled vectors i.e. we get $(u s_1 + v s_2, u v_1 + v v_2)$.
So far this behaves like just a vector, but here is where things are different.
Let us say now that instead I do $s_1 ^ {s_2}$. Is there a clear analogue of the exponential so that one can do $v_1^{v_2}$ and get a vector of the same dimension out of that? And similarly for other operations like logarithms, division. |
the immediate option is to just apply operations per component. But that doesn't seem correct, for example. Let us say our vector represents a color, let us make one object black the other white. If we take the components of the second to the power of the first we get a vector of all $1$'s, so white.
However, if $s_1$ is say, $5$ and $s_2$ is $-10$, doing $5^{-10}$ yields a value close to $0$, what happened to the scalars is very different to what happened to the colors.
I understand this sounds convoluted, it's for artistic purposes which is why I don;t have a strong mathemtical sense of what this should be.
I am wondering if people know of some kind of algebra that would study something like this.