Proof of R being a Vector Space - Best Resources Available

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Discussion Overview

The discussion revolves around the proof of the real numbers (R) being a vector space, including inquiries about resources and foundational concepts related to vector spaces. The scope includes theoretical aspects of vector spaces and the importance of specifying the base field in such discussions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant requests proof or resources regarding R as a vector space.
  • Another participant indicates they can prove it themselves and suggests closing the thread.
  • Some participants emphasize the importance of specifying the base field when discussing vector spaces, noting that R can be viewed over different fields, such as the complex numbers.
  • A participant explains that the definition of a vector space includes the requirement of being "over" a given field, which is essential for understanding scalar multiplication.
  • Clarifications are sought regarding what is meant by specifying the base field in the context of vector spaces.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of specifying the base field, with some emphasizing its importance while others focus on the proof aspect. The discussion does not reach a consensus on the best approach or resources for proving R as a vector space.

Contextual Notes

The discussion highlights the potential ambiguity in foundational definitions and the role of different fields in vector space theory, but does not resolve these issues.

woundedtiger4
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Hi everyone!

Can someone please tell me what is the proof of it? or Where can I find it?

Thanks in advance.

Best regards.
 
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Dear admin,

Please remove/close this thread as I can prove it myself :)

Thanks anyway.
 
Don't mean to nitpick , but it may be a good idea to specify the base field when you do these foundational exercises, i.e. the Reals are a vector space over 1-field? Clearly here there are not many options, but for R^2, the base field may be R or the Complexes (or maybe something else).
 
WWGD said:
Don't mean to nitpick , but it may be a good idea to specify the base field when you do these foundational exercises, i.e. the Reals are a vector space over 1-field? Clearly here there are not many options, but for R^2, the base field may be R or the Complexes (or maybe something else).
Many thanks for the reply. Can you please tell me that what do you mean by specifying the base field?
 
woundedtiger4 said:
Many thanks for the reply. Can you please tell me that what do you mean by specifying the base field?

Sure; when you say V is a vector space, you assume there is a set S of objects called vectors and a specific base field.
For example, the set ## \mathbb R^2 ## seen as the vectors can be made into a vector space either over the field of Complex numbers,or over the set of Real numbers (or over any field F with ## \mathbb R \subset F \subset \mathbb C ##). Look at the 4 bottom axioms of vector spaces in e.g., http://en.wikipedia.org/wiki/Vector_space that specify how the field properties relate to the vectors and their resp. properties.
 
Last edited:
The definition of "vector space" requires that it be "over" a given field. That is, one of the operations for a vector space is "scalar multiplication" in which a vector is multiplied by a member of the "base field". That field is typically the "rational numbers" (though rare), the "real numbers", or the "complex numbers".

Any field can be thought of as a one-dimensional vector space over itself.
 
Ivy ,who/what are you replying to?
 
I was responding to post #4 by WoundedTiger4: "Can you please tell me that what do you mean by specifying the base field?"
 

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