Talk:Linear combination

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It would be nice if someone wrote little about the history of the Linear combination - why it was invented in the first place / why there was the need for an object like a Linear combination in the first place. — Preceding unsigned comment added by Trismarck (talk • contribs) 01:59, 26 July 2012 (UTC)Reply

Hi All,

This is a request for some direction, from a non-mathematician. I plan to introduce a proof from linear algebra into the arbitrage pricing theory article. Basically, this is how the derivation there usually goes (where the generic-vectors below have a financial meaning):

If the fact that (1) a vector is orthogonal to n-1 vectors, implies that (2) it is also orthogonal to an nth vector, then (3) this nth vector can be formed as a linear combination of the other n-1 vectors.

Firstly, I would like to tighten up the wording such that it is acceptable; secondly, I would like to link the argument to the appropriate linear algebra theorem. Hope that's do-able.

Thanks. Fintor 13:25, 23 October 2006 (UTC)Reply

this article is balls. ok so you define an LC by explaining how its constructed, but wtf is it in plain terms? to me its just a pain in the arse —Preceding unsigned comment added by 84.67.47.10 (talk) 03:14, 11 December 2007 (UTC)Reply

I agree. This article is indeed balls. Round balls. Slightly squishy. We need a definition here. 157.127.124.14 (talk) 14:52, 4 September 2008 (UTC)Reply

In § Affine, conical, and convex combinations, the table giving various restrictions on combinations has useful info, but could be formatted better. In particular, it needs vertical rules to separate the columns more clearly. Perhaps we could use the wikitable template? I'll look into it if I can find time soon. yoyo (talk) 16:53, 23 March 2018 (UTC)Reply

Nope, it was the wikitable [CSS] class I was looking for. Anyway - Done! yoyo (talk) 17:19, 23 March 2018 (UTC)Reply

"Superposition" originally (and usually still?) refers to two things (e.g. geometric objects) being directly overlaid one on the other – in the case of shapes in an abstract geometric space, being made to coincide. I'm not finding a lot of sources which use the name "superposition" to refer to linear combinations. Is there a reliable source discussing this supposed terminology? –jacobolus (t) 19:48, 30 September 2025 (UTC)Reply

It's the main use of "superposition" in quantum mechanics. A superposition of two state vectors is a linear combination of them with complex coefficients. More generally, any area of physics that uses linear, homogeneous differential equations will talk about the superposition of solutions to them. Whether superposition should redirect here is a different question, but I'd guess that that's what the creator of the redirect had in mind. Stepwise Continuous Dysfunction (talk) 21:23, 30 September 2025 (UTC)Reply

The name "superposition" doesn't mean "linear combination" in either context from what I can tell. What we do have is the superposition principle, which is something like a statement that a vector quantity expressed in a couple of different ways should be the same. –jacobolus (t) 23:10, 30 September 2025 (UTC)Reply

Any textbook on quantum mechanics will say that a linear combination ${\displaystyle a|\psi \rangle +b|\phi \rangle }$ is a superposition of the vectors ${\displaystyle |\psi \rangle }$ and ${\displaystyle |\phi \rangle }$. What they mean by the "superposition principle" is that any such linear combination of valid quantum states is again a valid quantum state. See, e.g., pages 215 and 252–253 of Cohen-Tannoudji, Diu and Laloë's Quantum Mechanics, volume 1. Or page 96 of Sakurai and Napolitano. Stepwise Continuous Dysfunction (talk) 00:43, 1 October 2025 (UTC)Reply

This seems like a highly confused retroactive re-definition of the "superposition" in "superposition principle". But I guess it's plausible that sources do that. –jacobolus (t) 02:27, 1 October 2025 (UTC)Reply

I'm not an expert, but from glancing around at older sources, it seems like this might have come about from the idea of a "superposition" of multiple scalar or vector fields (meaning, in that context, layering one over the top of the other), with the idea that the combination of several effects could be computed by summing the independent effects at each point in space. Or for example that the combined wave resulting from several independent waves (such as light, sound, or water waves) could be computed by considering each separately, and adding their effects. This is a model which can still be useful even when it isn't quite physically accurate. Another example, from beam theory: "When several loads (concentrated or distributed) act upon a beam the total S.F. or B.M. at any section can be found by adding the S.Fs. and the B.Ms, at that sectiondue to each load considered separately. This is known as the principle of superposition." –jacobolus (t) 02:50, 1 October 2025 (UTC)Reply