A vector space is a mathematical structure formed by a collection of objects called vectors. These vectors can be added together and multiplied by numbers, known as scalars. The operations must satisfy certain rules, such as associativity and distributivity, allowing for consistent manipulation of the vectors. In a vector space, there are specific elements called basis vectors that can be combined to create any vector in that space. Common examples of vector spaces include Euclidean spaces like R² and R³, where vectors represent points in two or three dimensions, respectively.