Question

Which subsets of P₂ are subspaces of P₂
A. {p(t)∣p'(t) is constant }

Answer 1

The subset {p(t)∣p'(t) is constant } is a subspace of P₂ because it contains the zero vector, is closed under vector addition, and is closed under scalar multiplication.

Step-by-step explanation:

To determine which subsets of P₂ are subspaces of P₂, we need to check if they satisfy the three properties of a subspace:

1. They contain the zero vector.
2. They are closed under vector addition.
3. They are closed under scalar multiplication.

For the given subset {p(t)∣p'(t) is constant}, let's check if it satisfies these properties:

1. The zero vector is the constant function p(t) = 0, and its derivative is a constant, so it is contained in the subset.
2. If p₁'(t) and p₂'(t) are constants, then the sum (p₁ + p₂)'(t) = p₁'(t) + p₂'(t) is also a constant. Therefore, the subset is closed under vector addition.
3. If p'(t) is a constant, then for any scalar c, the derivative of the scalar multiple cp(t) is c times the derivative of p(t), which is still a constant. Thus, the subset is closed under scalar multiplication.

Therefore, the subset {p(t)∣p'(t) is constant } is a subspace of P₂.