Is R a Vector Space with Defined Operations? | Homework Statement

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SUMMARY

The discussion confirms that the set of real numbers R, under the defined operations of scalar multiplication by \(\alpha x = \alpha x\) and vector addition as \(x \oplus y = \max(x,y)\), does not constitute a vector space. The reasoning provided highlights the absence of a zero vector, as for any real number \(k\), there exists another number \(k-1\), preventing the existence of a unique zero vector necessary for vector space properties. Therefore, R is definitively not a proper vector space with these operations.

PREREQUISITES
  • Understanding of vector space axioms
  • Familiarity with scalar multiplication and vector addition
  • Knowledge of the concept of a zero vector
  • Basic comprehension of real numbers and their properties
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  • Study the axioms of vector spaces in linear algebra
  • Explore examples of valid vector spaces and their operations
  • Learn about alternative definitions of vector addition
  • Investigate the implications of zero vectors in vector space theory
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Students of linear algebra, mathematicians exploring vector space theory, and educators teaching concepts of scalar multiplication and vector addition.

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Homework Statement



Let R denote the set of real numbers. Define scalar multiplication by \alpha x = \alpha x which is simply regular scalar multiplication, and vector addition is defined as x \oplus y = max(x,y). Is R a vector space with these operations?

Homework Equations



Some given above.

The Attempt at a Solution



There seems to be no zero vector to this equation as for any number k there exists another number k-1, so there is no single 0 vector for a vector space with the operations defined above. Is this reasoning correct?
 
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Yes, that's true. So what is your answer to the question?
 
Then it is not a proper vector space!

Thanks a lot.
 

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