Question

Determine if the given set is a subspace of P₂. Justify your answer.

The set of all polynomials of the form p(t) = at², where a is in R.
Choose the correct answer below.
OA. The set is a subspace of P₂. The set contains the zero vector of P₂, the set is closed under vector addition, and the set is closed under multiplication on the left by mx2 matrices where m is any positive integer.
OB. The set is a subspace of P₂. The set contains the zero vector of P₂, the set is closed under vector addition, and the set is closed under multiplication by scalars.
OC. The set is not a subspace of P₂. The set is not closed under multiplication by scalars when the scalar is not an integer.
OD. The set is not a subspace of P₂. The set does not contain the zero vector of P₂

Answer 1

OB. The set is a subspace of P₂. The set contains the zero vector of P₂, the set is closed under vector addition, and the set is closed under multiplication by scalars.

To show this, we need to verify the three conditions for a set to be a subspace:

The set contains the zero vector: The zero vector of P₂ is 0t² = 0, which is in the set since any real number multiplied by 0 is 0.

The set is closed under vector addition: Let p(t) = at² and q(t) = bt² be two polynomials in the set. Then p(t) + q(t) = (a +