Determine whether the following set equipped with the given operations is a vector space. If it...

 

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Determine whether the following set equipped with the given operations is a vector space. If it is not a vector space identify the vector space axioms that fail. The set V of all real-valued functions f defined everywhere on the real line and such that f(0) = 4, with the operations (f+ g)(x) = f(x) + g(x) (kT)(x) = kf(C) (A) V is not a vector space. Axioms 2-3 and 7-10 hold, but Axioms 1, 4, 5, and 6 fail to hold. (B) V is not a vector space. Axioms 1-5 and 9-10 hold, but Axioms 6, 7, 8 fail to hold. (C) Vis not a vector space. Axioms 2-5 and 7-10 hold, but Axioms 1 and 6 fail to hold. (D) Vis a vector space. (E) V is not a vector space. Axioms 1-3 and 6-10 hold, but Axioms 4 and 5 fail to hold. (F) V is not a vector space. Axioms 4-10 hold, but Axioms 1, 2, 3 fail to hold. Determine whether the following set equipped with the given operations is a vector space. If it is not a vector space identify the vector space axioms that fail. The set V of all real-valued functions f defined everywhere on the real line and such that f(0) = 4, with the operations (f+ g)(x) = f(x) + g(x) (kT)(x) = kf(C) (A) V is not a vector space. Axioms 2-3 and 7-10 hold, but Axioms 1, 4, 5, and 6 fail to hold. (B) V is not a vector space. Axioms 1-5 and 9-10 hold, but Axioms 6, 7, 8 fail to hold. (C) Vis not a vector space. Axioms 2-5 and 7-10 hold, but Axioms 1 and 6 fail to hold. (D) Vis a vector space. (E) V is not a vector space. Axioms 1-3 and 6-10 hold, but Axioms 4 and 5 fail to hold. (F) V is not a vector space. Axioms 4-10 hold, but Axioms 1, 2, 3 fail to hold.

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