Insights Why Vector Spaces Explain The World: A Historical Perspective

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Vector spaces are defined as an abelian group combined with a field that operates on it, often visualized as arrows that can be manipulated through addition and scaling. The discussion emphasizes the abstract nature of vector spaces, allowing for the addition of various mathematical objects beyond simple arrows. Historical context is provided, highlighting significant contributions from figures like Schrödinger and Heisenberg in quantum mechanics. The article aims to connect the concept of vector spaces with their historical developments rather than provide a technical explanation. This exploration underscores the broader implications of vector spaces in understanding complex mathematical and physical theories.
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A vector space is an additively written abelian group together with a field that operates on it.​

Vector spaces are often described as a set of arrows, i.e. a line segment with a direction that can be added, stretched, or compressed. That’s where the term linear to describe addition and operation, and the term scalar for the scaling factor from the operating field come from. Although there is basically no difference between the two definitions, the abstract definition is preferable. Simply because we can add objects like sequences, power series, matrices or more general functions that are usually not associated with arrows, and we can have fields like finite fields, function fields, or p-adic numbers that are usually not considered to represent a stretching factor. ...

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This article does not aim to explain vector spaces. Its goal is to connect the concept and the historical developments. I have found and cited some interesting comments during my research (and included links to the original papers as far as it was possible without getting into conflicts with copyright laws), especially Schrödinger's remarks about the comparison of his formalism of QM with Heisenberg's. And where that "eigen-" thing came from.
 
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