Linear combinations, linearly independant famillies, generating famillies, bases, dimension
Définition : Linear combinations
Let
be a familly of
vectors of a
-vector space
We call linear combination of these vectors any vector of type
belonging to
The scalars
are called the coefficients of the linear combination.
Définition : Linearly independent famillies
A familly
of vectors of a vector space
is linearly independent if the only linear combination of these vectors equal to the zero vector is the one whose coefficients are all zero.
We also say that vectors
are linearly independent.
This can be expressed as:
is a linearly independent familly) is equivalent to:
Définition : Linearly dependent famillies
A non linearly independent familly is called a linearly dependent familly.
We also say that vectors
are linearly dependents.
This can be expressed as:
is a linearly dependent familly is equivalent to
Définition : Generating famillies
A familly
is generating a vector space
if any vector of
is equal to a linear combination of vectors of the familly, meaning:
Définition : Bases
A basis of a vector space is linearly independent generating familly.
