Math of Reference Frames: Spanning Vector Space

[kent davidge]

I'm aware that there are definitions of how reference frames translates to mathematics. But I've came to the following.

How incomplete would be to say that, mathematically speaking, two Lorentz (or whatever) inertial frames are two subspaces of a given vector space whose span is the same vector space? I mean

Let two "Lorentz frames" be $A$ and $B$, subsets of a set $V$ which is a vector space.

Then $\text{span} (A) = \text{span} (B) = V$.

What would be left by this reasoning?

 

[kent davidge]

Note that with this reasoning, we have all the carachteristics we have. For instance a four-vector is the same in two frames. In our words, it's the same in two subspaces... etc.

 

[PeterDonis]

How incomplete would be to say that, mathematically speaking, two Lorentz (or whatever) inertial frames are two subspaces of a given vector space whose span is the same vector space?

I think by the term "Lorentz inertial frames" you actually mean "sets of 4 basis vectors", correct? If that is the case, then yes, any Lorentz inertial frame is a set of basis vectors for the vector space of all 4-vectors. Basis vectors span the vector space by definition.