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 polynomials of even degree.

(1 pt) Does the following set of vectors constitute a vector space? Assume "standard" definitions of the operations The set of all polynomials of even degree. A. Yes O B. No If not, which condition(s) below does it fail? (Check all that apply) A. Vector spaces must be closed under addition B. Vector spaces must be closed under scalar multiplication C. There must be a zero vector D. Every vector must have an additive inverse LO E. Addition must be associative F. Addition must be commutative G. Scalar multiplication by 1 is the identity operation H. The distributive property 1. Scalar multiplication must be associative J. None of the above, it is a vector space

Explanation / Answer

The set of P (say) all polynomials of even degree is a vector space. The answer is Yes.

The sum of any 2 polynomials of an even degree is a polynomial of an even degree so that P is closed under vector addition.

The scalar multiple of any polynomial of an even degree is a polynomial of an even degree so that P is closed under scalar multiplication.

The zero polynomial is of degree 0, i.e. a polynomial of an even degree. Thus P contains 0.

The additive inverse of a0+a1x+a2x2 +…+an xn , where n is even, is - a0-a1x-a2x2 -…-an xn, which being a polynomial of an even degree , is in P.

All the other vector space axioms also hold.

Related Questions

Navigate

• Browse (All)
• Browse D
• Subjects

---

---

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