In each part of this question determine whether W is a subsp

In each part of this question, determine whether W is a subspace of the vector space V. Give a complete proof using the subspace theorem, or else give a specific example to show that some subspace property fails. Let V = F([a, b], R), i e. V contains all real valued functions on the interval [a, b] Let W = {f isin F([a, b], R)|f(6) = 10} where a lessthanorequalto 6 lessthanorequalto b. Let V = P_3, and let W = {a_3 x^3 + a_2 x^2 + a_1 x^1 + a_0|a_2 = 0} be the set of cubic polynomials with no quadratic factor. Let V = C the complex numbers viewed as a vector space over C, and let W be the subset of real numbers.

Solution

(a). Let f and g be 2 arbitrary functions in W and let k be an arbitrary scalar. Then f(6)= 0 and g(6) = 0 so that (f+g)(6) = f(6)+g(6) = 0+0 = 0. Hence (f+g) W, so that W is closed under vector addition. Further (kf)(6) = k(f(6)= k*0 = 0, hence, kf W, so that so that W is also closed under scalar multiplication. Further, apparently the zero function 0 W. Hence W is a vector space, and, therefore, a subspace of V.

(b). Let P(x) = a₃x³ +a₁x +a₀ and Q(x) = b₃x³ +b₁x +b₀ be 2 arbitrary elements of W and let k be an arbitrary scalar. Then P(x) +Q(x) = (a₃x³+a₁x+a₀)+(b₃x³+b₁x+b₀)=(a₃+b₃)x³+(a₁+b₁)x+(a₀+b₀). Since P(x) +Q(x) does not contain any quadratic factor, hence P(x)+Q(x)W. Also, kP(x)=k(a₃x³ +a₁x +a₀)=(ka₃x³+ka₁x+ka₀). Since kP(x) does not contain any quadratic factor, hence kP(x)W so that W is also closed under scalar multiplication. Further, apparently the zero polynomial 0 Was it does not contain any quadratic factor. Hence W is a vector space, and, therefore, a subspace of V= P₃.

(c ). Let a,b be any 2 arbitrary real numbers in W and let k be arbitrary scalar. Then a+b , being a real number is in W so that W is closed under vector addition. However, when k is a complex scalar, then         ka W. Therefore, W is not closed under scalar multiplication. Hence W is not a vector space and hence, W is not a subspace of V= C.