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 $\left({G, +_G, \circ}\right)_K$ 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 $\left({K, +_K, \times_K}\right)$ is by definition a field.

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

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 $\R^n$.

Also see

 * Scalar Field