Definition – Vector Space. – Math-reference.net

(!) Relates to definition of a field.

Definition

Let $F$ be a field with operations $+_1,\times_1$ , and let $V$ be a nonempty set. If $+_2,\times_2$ are operations, not necessarily the same as $+_1,\times_1$ , then we say that $V$ is a vector space (over $F$ ), and the elements of $V$ are called vectors, if the following are true,

To every , there is a corresponding element called the sum of and , with the following properties,
1. The $+_2$ operator is commutative: $\mathbf{x}+_2\mathbf{y}=\mathbf{y}+_2\mathbf{x}$ .
2. The operator $+_2$ is associative: $\mathbf{x}+_2(\mathbf{y}+_2\mathbf{z}) = (\mathbf{x}+_2 \mathbf{y})+_2 \mathbf{z}$ .
3. There exists a unique element $\mathbf{0}\in{V}$ , called zero vector, such that: $\mathbf{x}+_2\mathbf{0} = \mathbf{x}$ .
4. For every $\mathbf{x}\in{V}$ there is a correspdonding element $-\mathbf{x}\in{A}$ such that: $\mathbf{x}+_2(-\mathbf{x}) = \mathbf{0}$ .
For every , , there is an associated element , called the product of and with the following properties,
1. The operator $\times_2$ is associative: $(\alpha\times_{1}\beta)\times_{2}\mathbf{x} = \alpha\times_{2}(\beta\times_{2}\mathbf{x})$ .
2. $1\times_{2}\mathbf{x}=\mathbf{x}$ for $1\in{F}$ (where the $1$ is the one element of the field $F$ ).
3. The $\times_2$ operator is distributive: $\alpha\times_{2}(\mathbf{x}+\mathbf{y})=\alpha\times_{2}\mathbf{x}+\alpha\times_{2}\mathbf{y}$ and $(\alpha +_1 \beta)\times_{2}{\mathbf{x}} = \alpha\times_2 \mathbf{x} + \beta\times_2 \mathbf{x}$ .

Notes

Do consider the reference to the different operators $+_1, \times_1$ and $+_2, \times_2$ . For example, the usual definition of $+_2$ for the vector space $\mathbb{R}^3$ is the following,

$\mathbf{x}+_2{\mathbf{y}}=\begin{bmatrix} x_1+_{1}y_1\\x_2+_{1}y_2\\x_3+_{1}y_3\end{bmatrix}$

Which shows that, actually, we are well acustom to the fact that the operators differ, as this is the case with the usual definition. Thus, for example, matrix addition and multiplication is so defined that, given the set of matrices of dimension $n\times{m}$ , they form a vector space over $\mathbb{R}$ .

Note also that the first set of conditions just ensures that the set $V$ is an abelian group under the operation $+_2$ .