Proving Vector Space of All Real Numbers

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SUMMARY

The set of all real numbers constitutes a vector space over the field of real numbers (R). To prove this, one must demonstrate that vector addition and scalar multiplication are defined on the set, satisfying the necessary conditions. Specifically, adding two real numbers results in a real number, and multiplying a real number by another real number also yields a real number. This confirms the existence of a zero vector, establishing that the real numbers form a vector space, which is one-dimensional over R and extends to any subfield of R.

PREREQUISITES
  • Understanding of vector spaces and their properties
  • Familiarity with scalar multiplication and vector addition
  • Knowledge of fields, specifically the field of real numbers (R)
  • Basic concepts of linear algebra
NEXT STEPS
  • Study the axioms of vector spaces in linear algebra
  • Explore the concept of subfields of real numbers
  • Learn about one-dimensional vector spaces and their implications
  • Investigate examples of vector spaces over different fields
USEFUL FOR

Students of mathematics, particularly those studying linear algebra, educators teaching vector space concepts, and anyone interested in the foundational properties of real numbers in vector spaces.

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i know that the set "all real numbers" make up a vector space, but how do you prove that it is so?
 
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I would think that you would have to show that vector addition and scalar multiplication are defined on the set, subject to conditions.

Look here:

http://www.math.niu.edu/~beachy/courses/240/vectorspace.html
 
Over what field...

add up two real numbers, get a real number, multiply a real number by a real number get a real number, has a zero vector, therefore it's a vector space, over R, obviously 1-dimensional.

Equally obviously it is therefore a vector space over any subfield of R, not necessarily 1-d.
 

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