Vector Space: Fifth-Degree Polynomials

[eyehategod]

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?

 

[Gokul43201]

How is the degree of a polynomial defined? Now, what is the additive identity, and is it a fifth-degree polynomial?

 

[m2003]

A vector space is a mathematical structure that satisfies a set of axioms, which are a set of rules that define how the elements in the space behave. In order for a set to be a vector space, it must satisfy all of the axioms.

In this case, the set of all fifth-degree polynomials with the standard operations satisfies all of the axioms except for 4, 5, and 6. These axioms state that for any vector u in the set, there must exist a zero vector (0) and an additive inverse (-u) that also belong to the set, and that any scalar multiple of a vector in the set must also belong to the set.

In the set of fifth-degree polynomials, there is no zero vector (a polynomial with all coefficients equal to 0) and there is no additive inverse for every polynomial. Additionally, not all scalar multiples of a polynomial will result in a fifth-degree polynomial. For example, if we multiply a fifth-degree polynomial by 2, the resulting polynomial will have a degree of 10, which does not belong to the set. Therefore, this set does not satisfy the axioms and is not a vector space.

It is important to note that the axioms must hold for all possible vectors and scalars in the set, not just for a specific example. In this case, the failure of axioms 4, 5, and 6 means that the set is not closed under addition and scalar multiplication, which is a fundamental property of a vector space.