Does the following set of vectors constitute a vector space? Assume \"standard\"

Question

Does the following set of vectors constitute a vector space? Assume "standard" definitions of the operations.

The set of all continuous functions f defined on the interval [0,1] such that f(0)=1.

Does the following set of vectors constitute a vector space? Assume "standard" definitions of the operations. The set of all continuous functions f defined on the interval [0,1] such that f(0)=1. O A. Yes B. No If not, which condition(S) below does it fail? (Check all that apply) A. Vector spaces must be closed under addition C. There must be a zero vector E. Addition must be associative G. Scalar multiplication by 1 is the identity operation l. Scalar multiplication must be associative B. Vector spaces must be closed under scalar multiplication D. Every vector must have an additive inverse F. Addition must be commutative H. The distributive property J. None of the above, it is a vector space

Explanation / Answer

The set of all continuous functions defined on the interval[0,1], such that f(0)= 1 is not a vector space. The answer is No.

If f and g are 2 functions in the given set, then (f+g)(0) = f(0)+g(0) = 1+1 = 2 1 so that the set is not closed under vector addition.

If f is a function in the given set and k is a scalar, then (kf)(0) = k*f(0) = k so that the set is not closed under scalar multiplication.

The zero function, which maps 0 to 0 is not in the set.

The additive inverse of any function in the set is not in the set as –f(0) cannot be equal to -1.

Thus, options A,B,C,D do not hold.

Related Questions

Navigate

• Browse (All)
• Browse D
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.