Prove that the standard basis vectors span R^2

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SUMMARY

The discussion centers on the concept of vector spaces, specifically addressing why R^2 is considered a vector space over the field of real numbers (R) rather than the field of rational numbers (Q). It is established that while standard basis vectors e_1 and e_2 (or i and j) span R^2, attempting to define a basis over Q leads to complications, including the necessity of the axiom of choice due to the uncountably infinite nature of such a basis. This distinction is crucial for understanding the properties and dimensions of vector spaces.

PREREQUISITES
  • Understanding of vector spaces and their properties
  • Familiarity with standard basis vectors in R^2
  • Knowledge of fields, specifically R and Q
  • Basic comprehension of the axiom of choice in set theory
NEXT STEPS
  • Study the properties of vector spaces over different fields
  • Learn about the implications of the axiom of choice in linear algebra
  • Explore the concept of basis and dimension in vector spaces
  • Investigate the differences between countable and uncountable sets
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Students of linear algebra, mathematicians exploring vector spaces, and educators teaching the fundamentals of vector spaces and basis concepts.

Mathematicsresear
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Homework Statement


I know how to approach this problem; however, I'm just confused as to why we consider that R^2 is a vector space over the field R, and not Q or any other field for this question?

Standard basis vectors: e_1, e_2 or i,j
 
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Mathematicsresear said:

Homework Statement


I know how to approach this problem; however, I'm just confused as to why we consider that R^2 is a vector space over the field R, and not Q or any other field for this question?

Standard basis vectors: e_1, e_2 or i,j

You will have difficulty writing down a basis of ##\mathbb{R}^2## over the field ##\mathbb{Q}##: such a basis is uncountably infinite and you need the axiom of choice to show it exists.
 

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