Question

Which of the following sets is a subspace of set of prime numbers P Subscript nℙn for an appropriate value of​ n? A. All polynomials of the form p​(t)equals=aplus+bt squaredt2​, where a and b are in set of real numbers Rℝ B. All polynomials of degree exactly​ 4, with real coefficients C. All polynomials of degree at most​ 4, with positive coefficients

Answer 1

Check the explanation

Explanation:

A. All polynomials of the form p(t) = a + bt2, where a and b are in: This means that A is closed under scalar mult and vector addition, and includes the zero vector.

B.All polynomials of degree exactly 4, with real coefficients: what this means is that under vector addition, B isn't closed, and it does not consist of the zero vector. What it consist of is just polynomials with degree exactly 4. Let f=x4+1f=x4+1 and let g=−x4g=−x4. Both are in B, but their sum is not, because it has degree 0.

C. All polynomials of degree at most 4, with positive coefficients: what this means is that C is not a subspace for the reason that the positive coefficients make zero vector impossible. The restriction there also makes C not closed under multiplication by the scalar −1.

So the answer is only A :D