Vector Space: Fifth-Degree Polynomials

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SUMMARY

The discussion centers on the determination of whether the set of all fifth-degree polynomials forms a vector space under standard polynomial operations. It concludes that axioms 1, 4, 5, and 6 fail, indicating that this set does not satisfy the requirements for a vector space. Specifically, the failure of these axioms relates to the closure properties and the existence of an additive identity within the set of fifth-degree polynomials.

PREREQUISITES
  • Understanding of vector space axioms
  • Knowledge of polynomial degree definitions
  • Familiarity with standard operations on polynomials
  • Basic concepts of linear algebra
NEXT STEPS
  • Study the properties of vector spaces in linear algebra
  • Learn about polynomial operations and their implications
  • Investigate the concept of additive identity in vector spaces
  • Explore examples of sets that do and do not form vector spaces
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Students of linear algebra, mathematics educators, and anyone interested in the properties of polynomial functions and vector spaces.

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15. Determine wheter the set is a vector space.
The set of all fifth-degree polynomials with the standard operations.
AXIOMS
1.u+v is in V
2.u+v=v+u
3.u+(v+w)=(u+v)+w
4.u+0=u
5.u+(-u)=0
6. cu is in V
7.c(u+v)=cu+cv
8.(c+d)u=cu+cd
9.c(du)=(cd)u
10.1(u)=u

the axioms that fail are 1,4,5, and 6. I don't know why 4,5,and 6 fail. Can anyone help me?
 
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How is the degree of a polynomial defined? Now, what is the additive identity, and is it a fifth-degree polynomial?
 

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