MHB Abstract algebra: i need examples of ...

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SUMMARY

This discussion focuses on providing examples of vector spaces over different fields. The examples include the vector space of all vectors of the form and over the real numbers, as well as the space of all polynomials of degree 3 or less over the real numbers. Additionally, it addresses the concept of having the same vector space over multiple fields, specifically citing the vector space of complex numbers over the rational numbers, real numbers, and complex numbers. The distinction between vector spaces and their underlying fields is emphasized.

PREREQUISITES
  • Understanding of vector spaces and their definitions
  • Familiarity with fields in abstract algebra
  • Knowledge of polynomial functions and their properties
  • Basic comprehension of complex and real numbers
NEXT STEPS
  • Study the properties of vector spaces over different fields
  • Explore the concept of bases and dimensions in vector spaces
  • Learn about field extensions and their implications in vector space theory
  • Investigate the relationship between polynomials and vector spaces
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Students of abstract algebra, mathematicians exploring vector space theory, and educators seeking examples for teaching vector spaces and fields.

nweissma
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please offer me examples of: a) 3 vector spaces over the same field; and b) the same vector space over 3 fields.
 
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The space of all vectors of tne form <a, b>, where a and b are real number, over the real numbers.
The space or all vectors of the form <a, b, c> over the real numbers.
The space of all polynomials of degree 3 or less over the real numbers.

Since the underlying field is part of the definition of a vector space, I'm not sure I would agree that you can have the same vector space over different fields.

However, if I were required to answer such a question (!), I would say the vector space of all complex numbers over the field of
the rational numbers
the real numbers
the complex numbers.
 

Thread 'Confusion about the Moyal-Weyl twist'

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