Rank

$Echelon form$

λ_{i} = M in {k ∣ a_{ik} \neq = 0}

A

is in Echelon form:

λ_{1} > λ_{2} > \dots \land \forall i (a_{i λ_{i}} = 1)

A

becomes in echelon form by repeating the transformation by elementary matrices

r ank f := d im (I m f)

r ank A := r ank f

r ank A

is a maximum number of linear independent in column vectors

a_{1}, ..., a_{n}

of

A

r ank A^{T} = r ank A

r ank A :

number of non zero vector of column vectors