What is *not* a Vector Function?

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Discussion Overview

The discussion revolves around the concept of vector functions and what constitutes a function that is not a vector function. Participants explore definitions, examples, and the nature of different types of functions, including covector functions and tensors.

Discussion Character

  • Exploratory, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant expresses surprise at the prevalence of vector functions in their experience and questions what does not qualify as a vector function.
  • Another participant suggests that the definition of a vector function depends on the codomain, proposing an example of a function with a codomain that cannot be interpreted as a vector space.
  • A similar point is reiterated by another participant, who also mentions a linear embedding related to a specific vector space.
  • There is a correction regarding the number of elements in a proposed vector space, with one participant noting that it has 9 elements instead of 6, indicating a careful choice of numbers.
  • A participant introduces the idea of covector functions and mentions the existence of various tensors as alternatives to vector functions.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the definition of vector functions, and multiple competing views regarding what constitutes a non-vector function remain present throughout the discussion.

Contextual Notes

Some assumptions about the definitions of vector functions and codomains are not fully explored, and there are unresolved mathematical details regarding the examples provided.

nweissma
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i never thought of it before! every function that I've encountered has been a 'vector function' .. so what is not a vector function?
 
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Depends on what exactly you call "vector function". If the codomain has to be a vector space, then simply take a set as codomain that cannot be interpreted as vector space:
f:{1} -> {1,2,3,4,5,6}, f(1)=1.
 
mfb said:
Depends on what exactly you call "vector function". If the codomain has to be a vector space, then simply take a set as codomain that cannot be interpreted as vector space:
f:{1} -> {1,2,3,4,5,6}, f(1)=1.
But here ##f## is basically equal to the linear embedding of ##\{0\}## into the two-dimensional vector space ##\mathbb{Z}_3^2## :biggrin:
 
That vector space has 9 elements, not 6. That number was carefully chosen.
 
mfb said:
That vector space has 9 elements, not 6. That number was carefully chosen.
Ooops ...
 
nweissma said:
i never thought of it before! every function that I've encountered has been a 'vector function' .. so what is not a vector function?
there can be a covector function
There are a lot of different tensors
 
Last edited:

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