Is The Set of Polynomials of the Form p(t) = at^3 is a Subspace of ℙ3

Determine if the given set is a subspace of ℙ3. Justify your answer:The set of all polynomials of the form p​(t)=at^3​, where a is in ℝ.

To determine if the given set is a subspace of ℙ3, we need to check if it satisfies the three conditions for a subspace:

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To determine if the given set is a subspace of ℙ3, we need to check if it satisfies the three conditions for a subspace:

1. The set is closed under addition:
Let p1(t) = a1t^3 and p2(t) = a2t^3 be two polynomials in the set. Their sum is p1(t) + p2(t) = a1t^3 + a2t^3 = (a1 + a2)t^3. Since a1 and a2 are real numbers, (a1 + a2) is also a real number. Therefore, the sum of any two polynomials in the set is still a polynomial of the form at^3, where a is in ℝ. Hence, the set is closed under addition.

2. The set is closed under scalar multiplication:
Let p(t) = at^3 be a polynomial in the set, and c be a scalar. The scalar multiple of p(t) is c * p(t) = c * (at^3) = (ca)t^3. Since ca is a real number (as c and a are real numbers), the scalar multiple of p(t) is still a polynomial of the form at^3, where a is in ℝ. Hence, the set is closed under scalar multiplication.

3. The set contains the zero vector:
The zero vector in ℙ3 is the polynomial 0(t) = 0. It can be written in the form 0(t) = 0t^3. Since 0 is a real number, the zero vector is a polynomial of the form at^3, where a is in ℝ. Hence, the set contains the zero vector.

Since the set satisfies all three conditions for a subspace, namely closure under addition, closure under scalar multiplication, and containing the zero vector, the set of all polynomials of the form p(t) = at^3, where a is in ℝ, is indeed a subspace of ℙ3.

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