Definition:Vector Space

Definition
Let $\struct {K, +_K, \times_K}$ be a division ring.

Let $\struct {G, +_G}$ be an abelian group.

Let $\struct {G, +_G, \circ}_K$ be a unitary $K$-module.

Then $\struct {G, +_G, \circ}_K$ is a vector space over $K$ or a $K$-vector space.

That is, a vector space is a unitary module whose scalar ring is a division ring.

If $\times_K$ is commutative, then $\struct {K, +_K, \times_K}$ is by definition a field.

In that case, the scalar ring of $\struct {G, +_G, \circ}_K$ is called the scalar field of $\struct {G, +_G, \circ}_K$.

Also known as
A vector space is also sometimes called a linear space, especially when discussing the real vector space $\R^n$.

Some go further and refer to a linear vector space

The notation $\struct {G, +_G, \circ, K}$ can also be seen for this concept.

Also defined as
Some sources insist that $\struct {K, +_K, \times_K}$ needs to be a field, not just a division ring, for this definition to be valid.

Also see

 * Definition:Scalar Field

As a vector space is also a unitary module, all the results which apply to modules, and to unitary modules, also apply to vector spaces.