A vector space is a mathematical structure formed by a collection of vectors, which can be added together and multiplied by scalars. Vectors can represent quantities with both magnitude and direction, such as forces or velocities. The operations of addition and scalar multiplication must satisfy certain properties, including commutativity, associativity, and the existence of an additive identity (the zero vector). Vector spaces are fundamental in various fields, including linear algebra, physics, and computer science. They provide a framework for solving systems of linear equations and are essential for understanding concepts like linear transformations and basis vectors.