The definition of vector spaces is based on the addition and scalar multiplications, which were used in the expressions ** u** +

A vector space is a set ** V**, its elements

- addition
+`u`:`v`×`V`→`V``V`- scalar multiplication
`c`:**u****R**×→`V``V`

satisfying the following properties

- commutative
+`u`=`v`+`v``u`- associative
- (
+`u`) +`v`=`w`+ (`u`+`v`)`w` - zero
- There is a special vector
**0**∈such that`V`+`u`**0**==`u`**0**+`u` - negative
- For any
∈`u`there is a vector -`V`such that`u`+ (-`u`) =`u`**0** - distributive
`c`(+`u`) =`v``c`+`u``c``v`- distributive
- (
`c`+`d`)=`u``c`+`u``d``u` - associative
- (
`cd`)=`u``c`(`d`)**u** - one
- 1
=`u``u`

The eight properties comes from the properties satisfied by the euclidean vectors. Moreover, the notation **R**×** V** →

Briefly speaking, a vector space is any set up in which addition and scalar multiplication can be done in a reasonable way (as measured by the eight properties).

Example The euclidean space **R**` ^{n}`, with the following operations

(`x`_{1}, `x`_{2}, ..., `x _{n}`) + (

satisfies the eight properties and is the primary example of vector spaces.

Example Let `F`(** X**) be all the real valued functions on a set

(`f` + `g`)(`t`) = `f`(`t`) + `g`(`t`),

(`cf`)(`t`) = `c`(`f`(`t`)),

`F`(** X**) is a vector space. In fact, two functions can also be multiplied together. However, this operation is not needed for the vector space.

By imposing additional conditions on functions, we get

`C`(**R**) = all continuous functions,

`C`^{1}(**R**) = all functions `f`(`t`) with continuous derivative `f`'(`t`),

`C ^{k}`(

We also use `C`^{0}(**R**) to denote `C`(**R**) and use `C`^{∞}(**R**) to denote functions with derivatives of any order. With the same addition and scalar multiplication, these are also vector spaces. The important thing to verify here is that if `f`(`t`) and `g`(`t`) satisfy the additional conditions (say, continuous), then `f`(`t`) + `g`(`t`), `cf`(`t`) also satisfy the additional conditions (also continuous). The verification follows from the usual theorems in calculus.

Example Another special collection of functions is polynomials. For any `n`, let ` P_{n}` be the collection of polynomials

`p`(`t`) = `a`_{0} + `a`_{1}`t` + `a`_{2}`t`^{2} + ... + `a _{n}t^{n}`,

of degree ≤ `n`. The usual addition and scalar multiplication

(`a`_{0} + `a`_{1}`t` + `a`_{2}`t`^{2} + ... + `a _{n}t^{n}`) + (

are consistent with the similar operations for general functions, and have the eight properties. Therefore ` P_{n}` is a vector space.

Example Given `m` and `n`, all the `m` by `n` matrices, together with the usual addition and scalar multiplication, form a vector space `M`(`m`, `n`). The special collection `Sym`(`n`) of symmetric matrices, with the same operations, is also a vector space. Similar to the case of special collections of functions, the key reason for `Sym`(`n`) to be a vector space is ** A** and

Note that the additions and scalar multiplications in the examples have to be the routine one because otherwise some of the eight properties will fail.

Finally, in a vector space we may combine addition and scalar multiplication and form the linear combination `c`_{1}`u`_{1} + `c`_{2}`u`_{2} + ... + `c _{k}u_{k}`.