Insights Why Vector Spaces Explain The World: A Historical Perspective

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The discussion centers on the concept of vector spaces, defined as an additively written abelian group paired with a field that operates on it. It emphasizes the abstraction of vector spaces beyond simple geometric interpretations, allowing for the addition of various mathematical objects such as sequences and matrices. The historical context is highlighted, particularly referencing Schrödinger's and Heisenberg's contributions to quantum mechanics and their relation to vector space formalism. This exploration aims to connect mathematical concepts with their historical developments rather than provide a technical explanation of vector spaces.

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  • Understanding of abelian groups
  • Familiarity with fields in mathematics
  • Basic knowledge of linear algebra
  • Awareness of quantum mechanics principles
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  • Research the properties of abelian groups in depth
  • Study the applications of vector spaces in quantum mechanics
  • Explore the historical development of linear algebra concepts
  • Investigate the significance of eigenvalues and eigenvectors in various mathematical contexts
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Mathematicians, physicists, and students interested in the historical context of vector spaces and their applications in quantum mechanics.

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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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