Is R a Finite-Dimensional Vector Space Over Q?

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Discussion Overview

The discussion revolves around whether the set of all real numbers, R, is a finite-dimensional vector space over the field of rational numbers, Q. Participants explore the implications of cardinality and the nature of vector spaces in this context, referencing concepts from linear algebra and set theory.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that R is not a finite-dimensional vector space over Q, citing that if it were, it would imply that R is countable, which it is not.
  • One participant argues that R is not even countably-dimensional, suggesting that the dimension must be uncountable based on similar reasoning.
  • A request for a quick proof of the uncountable dimension of R over Q is made, with a suggestion to construct a Hamel basis to demonstrate this.
  • Another participant discusses the implications of the cardinality of sets formed by linear combinations of basis elements, indicating that if the index set for the basis is infinite, the dimension must be uncountable.
  • Some participants engage in a technical discussion about the cardinality of finite subsets of a basis and the implications for the dimension of R.
  • There is a mention of Schauder bases and a clarification that they are countable by definition, which contrasts with the discussion of Hamel bases.

Areas of Agreement / Disagreement

Participants generally agree that R is not a finite-dimensional vector space over Q. However, there is ongoing debate regarding the nature of its dimension, with some asserting it is uncountable and others discussing the implications of various mathematical constructs.

Contextual Notes

Participants reference the need for rigorous definitions and assumptions regarding cardinality and the nature of vector spaces, particularly in the context of infinite sets and bases. The discussion includes various mathematical arguments that are not fully resolved.

  • #31
something much neater!

If a vector space V is finite dimensional , of dim n say, over a field F
then, (assume, {v1,v2,...,vn} is a basis for V)
V is isomorphic to F^n
through the correspondence
a1.v1 +a2v2 + a3v3 + ... + anvn -----> (a1,a2,a3,...,an) {the n component tuple of F^n) (verify the bijective linear transformation)
It follows that R (reals) would be isomorphic to Q^n (Q denotes field of rationals)
But, Q^n is countable (Q being countable)
implying that R is countable (absurd)
Q.E.D.
 
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  • #32
sihag said:
something much neater!

If a vector space V is finite dimensional , of dim n say, over a field F
then, (assume, {v1,v2,...,vn} is a basis for V)
V is isomorphic to F^n
through the correspondence
a1.v1 +a2v2 + a3v3 + ... + anvn -----> (a1,a2,a3,...,an) {the n component tuple of F^n) (verify the bijective linear transformation)
It follows that R (reals) would be isomorphic to Q^n (Q denotes field of rationals)
But, Q^n is countable (Q being countable)
implying that R is countable (absurd)
Q.E.D.
How is this neater? It's exactly what HallsofIvy had in post #2.
 
  • #33
oops, i missed that ! : )
 
  • #34
And what could be neater than to quote me?
 

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