Post #3190601

2026-04-20 15:25 UTC

@simontatham@hachyderm.io I don't know if this goes anywhere, but in linear algebra, linear (in)dependence is a general notion, while orthogonality requires the additional structure of an inner product. Is there maybe some way of using an inner product to give some kind of quantitative notion of 'desirability'? I don't know if this can be made sane in the software world, but it would be interesting if it could.

Replies (1)

@simontatham@hachyderm.io 2026-04-20 16:26

@RobJLow@mathstodon.xyz hmmm, I think this ties in somewhat to @tuftyindigo@meow.social's subthread about relative lengths of the basis vectors. The property I described – that if you know e₁ out of some orthogonal set then you can recover the e₁ component of some other vector without needing to know all the other {eᵢ} – doesn't require the basis to be _orthonormal_, only _orthogonal_. If e₁ doesn't have unit length, then the formula is slightly more complicated, but still exists. So, in particular, that property is preserved if you apply a (non-singular) diagonal matrix transformation to the whole space, which scales each basis vector by an independently chosen factor. And that means there _can't_ be a way to recover the relative 'importance' (length) of the basis vectors given only the primitive of decomposing a vector into a component parallel to each one, because the relative lengths can change without changing that primitive. This suggests to me that it might be possible to define a weaker notion of an orthogonal basis which doesn't depend on a full-on inner product, because it corresponds to a whole equivalence class of orthogonal bases by the usual definition (those which are diagonal rescalings of each other). There's probably some set of axioms that characterises this notion without committing to a specific element of that equivalence class, in the same way that matroids characterise the notion of linear independence even after you throw away the arithmetic in the vector space.

Open ##3190602