On basic idea of random variable 2

Question

A while ago I asked the following question, On basic idea of a random variable.

Looking into probability theory again after some time made me think of the following analogy.

Consider the idea of an operator on a vector space, it has a matrix given some basis and without a basis the operator is rather abstract.

Is the situation the same with "random quantities"? I.e it is not a function unless you have a given probability space? without it it is just its distrubution in some sense.

Answer 1

Quote from @Did's comment above:

Most linear maps have several matrix representations (obtained by changing the basis of the source and target vector spaces defining the map). Any distribution can be realized by several random variables (obtained by changing the probability space and even, most often, the function defined on it). If this is the analogy you have in mind, then it is correct.