Question

Which of the following subsets of P₂ are subspaces of P₂?

A. p'(7) = p(8)

B. p(-t) = -p(t) for all t

Answer 1

To determine which of the given subsets of P₂ are subspaces, we need to check if they satisfy the three conditions for subspaces. Subset A does not satisfy the conditions, while subset B does.

Step-by-step explanation:

To determine which of the given subsets of P₂ are subspaces, we need to check if they satisfy the three conditions for subspaces:

1. The subset must contain the zero vector.
2. The subset must be closed under addition.
3. The subset must be closed under scalar multiplication.

Let's analyze each subset:

A. p(t)

• This subset does not contain the zero vector, as there is no polynomial that satisfies the condition p'(7) = p(8) and also p(0) = 0.
• This subset is not closed under addition, as the sum of two polynomials that satisfy the condition p'(7) = p(8) may not satisfy the condition.
• This subset is not closed under scalar multiplication, as multiplying a polynomial that satisfies the condition by a scalar may not satisfy the condition anymore.

Therefore, subset A is not a subspace of P₂.

B. p(-t) = -p(t) for all t

• This subset contains the zero vector, as the zero polynomial satisfies the condition p(-t) = -p(t) for all t.
• This subset is closed under addition, as the sum of two polynomials that satisfy the condition will also satisfy the condition.
• This subset is closed under scalar multiplication, as multiplying a polynomial that satisfies the condition by a scalar will still satisfy the condition.

Therefore, subset B is a subspace of P₂.