Exercice1
#1)

def g(v):
    l=v.list()
    return vector(QQ,[2*l[0]+l[1],l[1]+(1/2*l[2]),(3/2*l[0])+l[1]-5*l[2]])
u=vector(QQ,[1,1,1])
show(g(u))

#2)

e1=vector(QQ,[1,0,0])
e2=vector(QQ,[0,1,0])
e3=vector(QQ,[0,0,1])
M=matrix(QQ,[g(e1),g(e2),g(e3)])
M1=M.transpose()
print(M1)

#3)

u1=vector(QQ,[1,0,1])
u2=vector(QQ,[1,1,1])
u3=vector(QQ,[0,0,1])
M2=matrix(QQ,[u1,u2,u3])
print("si le determinant de M2!=0 alors u une base de Q3")
show(M2.determinant()!=0)

#4)
E=VectorSpace(QQ,3)
Eu=E.subspace_with_basis([u1,u2,u3])
mgl=matrix(QQ,[Eu.coordinates(g(u1)),Eu.coordinates(g(u2)),Eu.coordinates(g(u3))])
mgl1=mgl.transpose()
show(mgl1)
show(g(u1+u2+u3))