Definitely.kq6up said:I think there is a closure problem ##\ldots##
A "Not a Vector Space" is a set of objects that do not satisfy the axioms and properties of a vector space. In other words, it is a collection of elements that do not follow the rules of vector addition and scalar multiplication.
The axioms of a vector space are a set of rules that must be satisfied in order for a set of objects to be considered a vector space. These include closure under addition and scalar multiplication, associativity, commutativity, distributivity, existence of a zero vector, and existence of additive inverses.
Understanding "Not a Vector Space" is important because it helps us identify and distinguish between sets that are and are not vector spaces. This knowledge can also help us identify where the axioms and properties of vector spaces break down, providing insight into the limitations of vector spaces and their applications.
No, a set cannot be partially a vector space. A set must satisfy all of the axioms and properties of a vector space in order to be considered a vector space. If a set does not satisfy even one of the axioms, it is not a vector space.
Some examples of "Not a Vector Space" include the set of all real numbers with the operation of multiplication, the set of all integers with the operation of division, and the set of all polynomials with the operation of composition. These sets do not satisfy the axioms and properties of a vector space, making them "Not a Vector Space".