Question

Determine whether the set, together with the indicated operations, is a vector space. If it is not, then identify one of the vector space axioms that fails. The set of all eighth-degree polynomials with the standard operations.
(A) The set is a vector space.
(B) The set is not a vector space because it is not closed under addition.
(C) The set is not a vector space because the associative property of addition is not satisfied
(D) The set is not a vector space because an additive inverse does not exist.
(E) The set is not a vector space because the distributive property of scalar multiplication is not satisfied.

Answer 1

Answer:

The answer to the question are

(B) The set is not a vector space because it is not closed under addition. and

(D) The set is not a vector space because an additive inverse does not exist.

Step-by-step explanation:

To be able to identify the possible things that can affect a possible vector space one would have to practice on several exercises.

The vector space axioms that failed are as follows

(B) The set is not a vector space because it is not closed under addition.

(2·x⁸ + 3·x) + (-2·x⁸ +x) = 4·x which is not an eighth degree polynomial

(D) The set is not a vector space because an additive inverse does not exist.

There is no eight degree polynomial  = 0

The axioms for real vector space are

• Addition: Possibility of forming the sum  x+y which is in X from elements  x and y which are also in X
• Inverse: Possibility of forming an inverse -x which is in X from an element x which is in X
• Scalar multiplication: The possibility of forming multiplication between an element x in X and a real number c of which the product cx is also an element of X