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 (bv)
• 1v = v
• (a + b)v = av + bv
• a(v + w) =av + aw

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

a1v1 + a2v2 + ... anvn

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