# Vector Space

## Set of elements called vectors, with 2 operations with certain properties:

Usually we will note a vector in boldfaced letters or
with an arrow above it

### 1 ) Addition of vectors : commutative, associative, with an identity element
and each element must have inverses

**u** + **v** = **v** + **u**
**u** + ( **v** + **w** ) = (**u** + **v**
) + **w**
- there is
**0** such as for any vector **v**, **0** + **v** =
**v**
- for every vector
**v**, there is another vector **w** such as **v**
+ **w** = **0** . **w** is written " **-v** "

### 2 ) Scalar Multiplication : associative and distributive

- (ab)
**v** = a (b**v**)
- 1
**v** = **v**
- (a + b)
**v** = a**v** + b**v**
- a(
**v** + **w**) =a**v** + a**w**

### Given those two operations, we may define a "linear combination"
of a set of vector **v1**, ... **vn** : any vector of the form

a1**v1** + a2**v2** + ... an**vn**

### Example of Vector Space : Rn (the set of all ordered n-tuples of real number)
is a vector space

### Sub example : R3 :

/ r1 / 1 / 4 / 5
| r2 ...... ex : | 3 + | 5 = |12
\ r3 \ 6 \ 2 \ 8