Determining Subspaces: Set of Polynomials with Positive Coefficients of Degree 8 or Less

Determine if the given set is a subspace of ℙ8. Justify your answer:All polynomials of degree at most 8​, with positive real numbers as coefficients.

To determine if a set is a subspace of a vector space, we need to check three properties: closure under addition, closure under scalar multiplication, and contains the zero vector

To determine if a set is a subspace of a vector space, we need to check three properties: closure under addition, closure under scalar multiplication, and contains the zero vector.

In this case, we are given the set of all polynomials of degree at most 8 with positive real numbers as coefficients. Let’s call this set S.

1. Closure under addition: For any two polynomials f(x) and g(x) in S, their sum f(x)+g(x) will also be a polynomial of degree at most 8. Additionally, since the coefficients of f(x) and g(x) are positive real numbers, the coefficients of f(x)+g(x) will also be positive real numbers. Therefore, S is closed under addition.

2. Closure under scalar multiplication: For any polynomial f(x) in S and any scalar c, the product c*f(x) will also be a polynomial of degree at most 8. Since multiplying positive real numbers by another positive real number gives a positive real number, the coefficients of c*f(x) will still be positive real numbers. Therefore, S is closed under scalar multiplication.

3. Contains the zero vector: The zero polynomial, which has all coefficients equal to zero, is a polynomial of degree at most 8. However, the zero polynomial does not have positive real numbers as coefficients. Therefore, S does not contain the zero vector.

Since S does not satisfy the third property, it is not a subspace of ℙ8.

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