Question

Determine if the given set is a subspace of Pₙ for an appropriate value of n.

A. Set of all polynomials with degree less than n
B. Set of all polynomials with positive coefficients
C. Set of all polynomials with odd degree
D. Set of all constant polynomials

Answer 1

A. Yes, B. No, C. Yes, D. Yes.

Step-by-step explanation:

A. The set of all polynomials with degree less than n is a subspace of Pₙ. This is because it satisfies the three conditions that define a subspace: closure under addition, closure under scalar multiplication, and contains the zero vector.

B. The set of all polynomials with positive coefficients is not a subspace of Pₙ. This is because it does not satisfy the closure under scalar multiplication condition. If we multiply a polynomial with positive coefficients by a negative scalar, the resulting polynomial will not have positive coefficients.

C. The set of all polynomials with odd degree is a subspace of Pₙ. It satisfies the three conditions and contains the zero polynomial, which has an odd degree.

D. The set of all constant polynomials is a subspace of Pₙ. It satisfies the three conditions and contains the zero polynomial, which is a constant polynomial.