A1.1. Definition of a vector space

Quantities , ,.., called vectors, belong to a vector space ( V) (also called linear space), if the following axioms are satisfied:

1. ? , ? ( V) ? + = + = ? ( V)

2. ? , , ? ( V) ? ( + ) + = + ( + ) = ? ( V)

3. There exists a unique null vector such that + = ? ? ( V)

4. ? ? ( V) there exists a unique vector noted , such that: + ( ) =

Moreover, to every vector and every number ?, there corresponds a vector ? such that:

1. ?( ? ) = ( ??)

2. ( ? + ?) = ? + ?

3. ?( + ) = ? + ?

4. 1 =

A1.2. Dependence and independence of vectors

is said to be to be a linear combination of N vectors { , ,... }, if there exist numbers { ? ₁, ? ₂,... ? _N} such that:

(A1.1)

Let { , ,... } be a set of N vectors and consider the linear combination:

(A1.2)

If equation [A1.2] has no other solution than ? ₁ = ? ₂ =,... =