Vector spaces are even more complex algebraic structures than fields. In fact, they consist of two algebraic structures combined together, an Abelian group of elements called vectors and a field, the elements of which are used to manipulate the vectors in an operation called the scalar multiplication. The scalar multiplication adds special, additional properties to the vector space, which are not available in its components - neither in the field nor in the Abelian group. A vivid description of such properties is that vectors can be "stretched" or "compressed" when multiplied by a scalar from the field.
This chapter provides a definition of vector spaces, explains what subspaces of vector spaces are, and provides some examples and basic conclusions from those definitions.