Question

Determine whether the given set S is a subspace of the vector space V.

A. V=P₃, and S is the subset of P₃ consisting of all polynomials of the form p(t) = a + bt + ct², where a, b, and c are constants and a + b + c = 0.

Answer 1

The set S is a subspace of vector space V since it satisfies the conditions of closure under addition and scalar multiplication, as the condition a + b + c = 0 is preserved.

Step-by-step explanation:

The student has asked whether a given set, S, is a subspace of the vector space V, where V is defined as P₃ (the space of all polynomials of degree at most 3), and set S consists of all polynomials of the form p(t) = a + bt + ct², with the condition that a + b + c = 0. To determine if S is a subspace, two conditions must be met: closure under addition and scalar multiplication.

Closure under addition means if p1(t) and p2(t) are in S, then their sum p1(t) + p2(t) must also be in S. Since a1 + b1 + c1 = 0 and a2 + b2 + c2 = 0, the sum will have coefficients (a1+a2) + (b1+b2)t + (c1+c2)t², and (a1+a2) + (b1+b2) + (c1+c2) = 0, ensuring closure under addition.

Closure under scalar multiplication implies that if p(t) is in S, then so is kp(t) for any scalar k. Since scalar multiplication k will just multiply each coefficient by k, the sum (ka) + (kb) + (kc) = k(a + b + c) = k(0) = 0, affirming closure under scalar multiplication.

Therefore, set S is indeed a subspace of the vector space V as it satisfies both subspace conditions.