Vector Space

Definition (Vector Space)

A vector space over a field $\mathbb{F}$ is a non-empty set $V$ together with a binary operation and a binary function that satisfy the following eight axioms.

The binary operation called vector addition or simply addition assigns to any two vectors $\mathbf{v},\mathbf{w}\in V$ a third vector $\mathbf{v}+\mathbf{w}\in V$. The binary function called scalar multiplication assigns to any scalar $a\in \mathbb{F}$ and any vector $\mathbf{v}\in V$ a vector $a\mathbf{v}\in V$.

Axiom	Description
Associativity of addition	$\forall \mathbf{u},\mathbf{v},\mathbf{w}\in V$, we have $(\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})$.
Commutativity of addition	$\forall \mathbf{u},\mathbf{v}\in V$, we have $\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}$.
Identity element of addition	$\exists 0\in V$ such that for all $\mathbf{v}\in V$, $\mathbf{v}+0=\mathbf{v}$.
Inverse elements of addition	$\forall \mathbf{v}\in V$, there exists an element $-\mathbf{v}\in V$ such that $\mathbf{v}+(-\mathbf{v})=0$.
Compatibility of scalar multiplication with field multiplication	$\forall \mathbf{a},\mathbf{b}\in \mathbb{F}$ and $\mathbf{v}\in V$, we have $\mathbf{a}(\mathbf{bv})=(\mathbf{ab})\mathbf{v}$.
Identity element of scalar multiplication	$\forall \mathbf{v}\in V$, we have $1\mathbf{v}=\mathbf{v}$, where $1$ is the multiplicative identity in $\mathbb{F}$.
Distributivity of scalar multiplication with respect to vector addition	$\forall \mathbf{a}\in \mathbb{F}$ and $\mathbf{u},\mathbf{v}\in V$, we have $\mathbf{a}(\mathbf{u}+\mathbf{v})=\mathbf{au}+\mathbf{av}$.
Distributivity of scalar multiplication with respect to field addition	$\forall \mathbf{a},\mathbf{b}\in \mathbb{F}$ and $\mathbf{v}\in V$, we have $(\mathbf{a}+\mathbf{b})\mathbf{v}=\mathbf{av}+\mathbf{bv}$.

Theorem (Basis)

Each vector space $V$ admits a basis ${\vec{e}}_1,{\vec{e}}_2,\dots ,{\vec{e}}_n\in V$ so that every $\vec{u}\in V$ can be uniquely expressed as a linear combination of the basis vectors. That is, there is a unique $n$-tuple of scalars $(u^1,u^2,\dots ,u^n)\in {\mathbb{F}}^n$ such that

$$\vec{u}=\sum _{i=1}^n{u}^i{\vec{e}}_i.$$

The number $n$ of vectors in a basis is independent of the choice of basis. $n$ is called the dimension of $V$, denoted $\dim V$. Note that it could be infinite.

In a plain vector space $V$, there is no distinguished choice for a basis. Only the “dimensionless” Cartesian space ${\mathbb{R}}^n=\mathbb{R}\times \cdots \times \mathbb{R}$ has a special basis ${\vec{e}}_1=(1,0,\dots ,0),{\vec{e}}_2=(0,1,\dots ,0),\dots ,{\vec{e}}_n=(0,0,\dots ,1)$ called the standard basis. In general, there is no canonical way to identify a vector space with ${\mathbb{R}}^n$ for some $n$.