Let 𝒞 ⊂ℝ^3 denote the set of vectors 𝐚=[ a1 a2 a3 ] such that the polynomial a1+

step 1 This time, S is the set of all polynomials of the form, A plus Bx cubed, plus C, x to the fourth

Let 𝒞 ⊂ℝ^3 denote the set of vectors 𝐚=[ a1 a2 ...

Let $\mathcal{C} \subset \mathbb{R}^3$ denote the set of vectors $\mathbf{a}=\left[\begin{array}{l}a_1 \\ a_2 \\ a_3\end{array}\right]$ such that the polynomial $a_1+a_2 t+a_3 t^2 \geq 0$ for every $t \in \mathbb{R}$. Show that it is closed under vector addition (i.e., $\mathbf{a}, \mathbf{b} \in \mathcal{C} \Longrightarrow \mathbf{a}+\mathbf{b} \in \mathcal{C}$ ) and under multiplication by positive numbers (i.e., $\mathbf{a} \in \mathcal{C}$ and $\alpha>0 \Longrightarrow \alpha \mathbf{a} \in \mathcal{C}$ ), but that $\mathcal{C}$ is not a vector space over $\mathbb{R}$. [REMARK: A set $\mathcal{C}$ with the indicated two properties is called a cone.]

Key Concepts

Cone

A cone in a vector space is a set that is closed under addition and under multiplication by positive scalars. This means that if two vectors belong to the cone, then their sum also belongs to it, and if a vector belongs to the cone, then multiplying it by any positive scalar produces another vector in the cone. However, a cone is not necessarily a vector space because it might not be closed under multiplication by negative scalars.

Vector Space

A vector space is a mathematical structure formed by a set of vectors along with operations of addition and scalar multiplication, which satisfy certain axioms such as associativity, distributivity, existence of an additive identity, and existence of additive inverses for every element. Closure under multiplication by all scalars, including negative numbers, is a requirement for a set to be a vector space.

Closure Properties

Closure properties refer to the idea that when you perform a specific operation on elements of a set, the result still belongs to that set. In the context of the problem, closure under vector addition and under multiplication by positive scalars confirms that the set is a cone, even though it lacks the full closure required for a vector space.

Nonnegative Polynomial

A nonnegative polynomial is a polynomial that yields a value greater than or equal to zero for every input in its domain. In the problem, the set is defined by this property, which imposes constraints on the coefficients of the polynomial and limits the set in such a way that, while it retains certain algebraic closure properties, it does not satisfy all the criteria necessary for a vector space.

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