Let n greaterthanorequalto 1 and V Pn be the vector space o

Let n greaterthanorequalto 1 and V = P_n be the vector space over R of real polynomials of degree less than or equal to n, i.e., those of the form p(t) - a_nt^n + a_n - 1 t^n - 1 + ... + a_1 t + a_0 Which a_0, a_1, ..., a_n epsilon R. Which of the following are subspaces of V? W_1 = {p(t) epsilon P_n: p(0) = p(1)} W_2 = Pp(t) epsilon p(0) - p(1) + p(2) = 3} W_3 = {p(t) epsilon p_n: p(0) - p\'(1) + p\"(2) =0}, where as usual p\', p\" denote first and second derivatives.

Solution

i)

1. 0 belongs to this set

2. Let , p ,q belong to this set

(p+q)(0)=p(0)+q(0)=p(1)+q(1)=(p+q)(1)

Hence, p+q belongs to the set

3. LEt c be a scalar and p belong to the set

(cp)(0)=cp(0)=cp(1)=(cp)(1)

Hence, cp belongs to the set

Hence the set is a subspace

ii)

Let, p,q belong to this set

(p+q)(0)-(p+q)(1)+(p+q)(2)=p(0)-p(1)+p(2)+q(0)-q(1)+q(2)=3+3=6

Hence, p+q does not belong to the set

HEnc set is not a vector space.

iii)

1. 0 belongs to this set

2. Let ,p,q belong to this set

(p+q)(0)-(p+q)\'(1)+(p+q)\'\'(2)=p(0)-p\'(1)+p\'\'(2)+q(0)-q\'(1)+q\'\'(2)=0+0=0

Hence, p+q belongs to the set

3. Let c be a scalar and p belong to the set

(cp)(0)-(cp)\'(1)+(cp)\'\'(2)=cp(0)-cp\'(1)+cp\'\'(2)=c(p(0)-p\'(1)+p\'\'(2))=c*0=0

HEnce, cp belongs the set

Hence, it is a subspace.