#Exercice 1



M=matrix(ZZ,[[2,2,3],[4,-1,7],[3,-2,5]])
M

M.nrows()
M.ncols()


N1=M;
N1.swap_rows(1,2);
M

N2=copy(M)
N2.swap_rows(1,2);
M






#Exercice 2


A=matrix(ZZ,[[3,6,9,9,15],[15,20,25,-10,10],[11,32,43,-10,22],[-6,0,1,-1,-1]])
A


A_H,u=A.hermite_form(transformation=True)
A_H
u
u*A

At=A.transpose()
AtH,v=At.hermite_form(transformation=True)

show(AtH.transpose(),v.transpose())

A*v.transpose()




# rg Imf=3 engendré par les vecteurs (3,0,0,-1),(0,5,1,-3),(0,0,2,1)

#Ker f est engendré par les deux dernières colonnes de v (18,0,98,131,-141),(0,1,0,1,-1)
A.right_kernel()



Y=vector([6,-5,-1,1])
A\Y
X=vector([4/5,-81/10,29/5,0,0])
A*X

I=matrix(ZZ,[[1,0],[0,1]])
w=vector([7,3])


w_second=w_prime
w_second
w_second[0],w_second[1]=w_second[1],w_second[0]
w_second

#Exercice 3
def matricedetransvection(q):
    return(matrix(ZZ,[[1,-q],[0,1]]))





def ReductionColonne2mieux(v):
    I=matrix(ZZ,[[1,0],[0,1]])
    J=matrix(ZZ,[[0,1],[1,0]])
    if v[1]==0:
        return(v,I)
    if v[0]==0:
        v[0],v[1]=v[1],v[0]
        return(v,J)
    if v[0]<0:
        v[0]=-v[0]
        I=I*matrix(ZZ,[[-1,0],[0,1]])
    if v[1]<0:
        v[1]=-v[1]
        I=I*matrix(ZZ,[[1,0],[0,-1]])
    if v[0]==v[1]:
        L=matrix(ZZ,[[1,0],[-1,1]])
        res=L*v
        return(L*I,res)
    if v[0]<v[1]:
        v[0],v[1]=v[1],v[0]
    t=v.coefficients()
    q=t[0]//t[1]
    r=t[0]%t[1]
    while r!=0:
        L=matricedetransvection(q)
        I=L*I
        v=L*v
        v[0],v[1]=v[1],v[0]
        I=J*I
        t=v.coefficients()
        q=t[0]//t[1]
        r=t[0]%t[1]
    L=matricedetransvection(q)
    v=L*v
    I=L*I
    v[0],v[1]=v[1],v[0]
    I=J*I
    return(v,I)