Let X be a set together with operations of addition, +:X×X→X, and multiplication, ⋅:R×X→X, (we write ax instead of a⋅x). \(X...Let X be a set together with operations of addition, +:X×X→X, and multiplication, ⋅:R×X→X, (we write ax instead of a⋅x). X is called a vector space (or a real vector space) if the following conditions are satisfied: (Addition is associative) If u,v,w∈X, then u+(v+w)=(u+v)+w. (Addition is commutative) If u,v∈X, then u+v=v+u. (Additive identity) There is a 0∈X such that v+0=v…