Question

Which of the following subsets of P_2 are subspaces of P_2?

A. \ \ p'(t) is constant \
B. \ \ p'(t) + 5 p(t) + 8 =0 \
C. \{ p(t) \ | \ \int_0^{4} p(t)dt = 0 \}
D. \ p(t) \
E. \ p(t) \
F. \ p(t) \

Answer 1

Subsets B and D are subspaces of P_2.

Step-by-step explanation:

Subset A: This subset is not a subspace of P_2 because the zero vector, which is the constant function 0, is not in the set. Therefore, it fails the subspace test for containing the zero vector.

Subset B: This subset is a subspace of P_2 because it satisfies all three conditions for being a subspace. The zero vector is in the set, the set is closed under addition and scalar multiplication, and it contains the additive inverse of any element in the set.

Subset C: This subset is not a subspace of P_2 because it fails the scalar multiplication test. The zero vector is not in the set and scalar multiplication does not keep an element in the set.

Subset D: This subset is a subspace of P_2 because it satisfies all three conditions for being a subspace.

Subset E: This subset is not a subspace of P_2 because it fails the scalar multiplication test. The zero vector is not in the set and scalar multiplication does not keep an element in the set.

Subset F: This subset is not a subspace of P_2 because it fails the scalar multiplication test. The zero vector is not in the set and scalar multiplication does not keep an element in the set.