Vector Space

A vector space over a field $F$ is a set $V$ equipped with

• an addition operation $+:V\times V\to V$, and
• a scalar multiplication operation $\cdot:F\times V\to V$,

such that for all $u,v,w\in V$ and all $a,b\in F$:

1. $u+(v+w)=(u+v)+w$ (associativity of $+$),
2. $u+v=v+u$ (commutativity of $+$),
3. there exists an element $0\in V$ with $v+0=v$ (additive identity),
4. for every $v\in V$ there exists an element $-v\in V$ with $v+(-v)=0$ (additive inverse),
5. $a\cdot(u+v)=a\cdot u+a\cdot v$ (distributivity over vector addition),
6. $(a+b)\cdot v=a\cdot v+b\cdot v$ (distributivity over scalar addition),
7. $(ab)\cdot v=a\cdot(b\cdot v)$ (compatibility with scalar multiplication),
8. $1\cdot v=v$, where $1$ is the multiplicative identity in $F$.

Vector spaces are the basic objects of linear algebra; linear maps are the structure-preserving functions between them. A prototypical family of examples is Euclidean space , where vectors are tuples of numbers.

Examples:

• For any $k\in\mathbb{N}$, the set $\mathbb{R}^k$ of $k$-tuples of real numbers is a vector space over $\mathbb{R}$ with coordinatewise addition and scalar multiplication.
• The polynomial ring $F[t]$ (viewed only with its addition and scalar multiplication by $F$) is a vector space over $F$.
• If $X$ is any set, then the set of all functions $X\to F$ is a vector space over $F$ under pointwise operations.