What does it mean if "V" is closed under "o-plus" and "o-dot"?

If a set "V" is said to be closed under "o-plus" (vector addition) and "o-dot" (scalar multiplication), it indicates that for any two elements (vectors) in "V," their sum is also an element of "V," and that multiplying any vector in "V" by a scalar results in another vector that is also in "V."

This closure property is a fundamental requirement for a set to qualify as a vector space. In fact, vector spaces must satisfy several axioms, including associativity, commutativity of addition, and the existence of additive inverses and identity elements. Since closure under addition and scalar multiplication is a critical requirement of these definitions, confirming that a set meets these conditions implies it is indeed a vector space.

In the context of vector spaces, the objects typically refer to vectors—quantities that have both magnitude and direction, which can exist in multiple dimensions. This distinguishes them from scalars, which are mere numerical values without directionality.

The choice indicating that "V" is a vector space and its objects are vectors accurately reflects these properties and definitions of vector spaces in linear algebra. Other options do not align with the concept of vector spaces as they either misconstrue the nature