Definition:Vector Space

Definition
Let $$\left({K, +_K, \times_K}\right)$$ be a division ring.

Let $$\left({G, +_G}\right)$$ be an abelian group.

Let $$\left({G, +_G, \circ}\right)$$ be a unitary $K$-module.

Then $$\left({G, +_G, \circ}\right)$$ 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.

Scalar Field
If $$\left({K, +_K, \times_K}\right)$$ is a field (i.e. $$\times_K$$ is commutative), then it is called the scalar field (of $$G$$), rather than scalar ring.

Vector
The elements of $$\left({G, +_G}\right)$$ are called vectors.

There exists a convention for annotating for a general vector in a style that distinguishes a vector from a scalar. This annotation varies, as follows.

Let $$x_1, x_2, \ldots, x_n$$ be a collection of scalars.

The vector $$\left({x_1, x_2, \ldots, x_n}\right)$$ can be annotated as:


 * $$\mathbf{x} = \left({x_1, x_2, \ldots, x_n}\right)$$
 * $$\vec{x} = \left({x_1, x_2, \ldots, x_n}\right)$$
 * $$\hat{x} = \left({x_1, x_2, \ldots, x_n}\right)$$
 * $$\underline{x} = \left({x_1, x_2, \ldots, x_n}\right)$$
 * $$\tilde{x} = \left({x_1, x_2, \ldots, x_n}\right)$$

In printed material the boldface $$\mathbf{x}$$ is common, but for handwritten material (where boldface is difficult to render) the "tilde" version $$\tilde{x}$$ or "arrow" version $$\vec{x}$$ are more usual. The "hat" version $$\hat{x}$$ usually has a more specialized meaning, namely to symbolize a unit vector.

Zero Vector
The identity of $$\left({G, +_G}\right)$$ is usually denoted $$\mathbf 0$$, or some variant of this, and called the zero vector.

Note that on occasion it is advantageous to denote the zero vector differently, for example by $$e$$, in order to highlight the fact that the zero vector is not the same object as the zero scalar.

Comment
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.

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