Respuesta :
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
The set is not vector space because,
(B). The set is not a vector space because it is not closed under addition.
(D). The set is not a vector space because an additive inverse does not exist.
Vector space :
A vector space is a set of objects called vectors, which may be added together and multiplied by numbers called scalars.
The vector space axioms are following,
- Addition: Possibility of forming the sum [tex]x+y[/tex] which is in[tex]X[/tex] from elements [tex]x[/tex] and [tex]y[/tex] which are also in [tex]X[/tex]
- Inverse: Possibility of forming an inverse [tex]-x[/tex] which is in [tex]X[/tex] from an element [tex]x[/tex] which is in [tex]X[/tex]
- Scalar multiplication: The possibility of forming multiplication between an element [tex]x[/tex] in [tex]X[/tex] and a real number [tex]c[/tex] of which the product [tex]cx[/tex] is also an element of[tex]X[/tex]
Learn more about the vector space here:
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