typedef vector < double >Trow;
typedef vector < Trow > Tmatrix;

template < class T > ostream &	// template to print any vector 
  operator << (ostream & os, vector < T > v)
{
  for (int i = 0; i < v.size (); i++)
    os << " " << v[i];
  os << "\n";
  return os;
}

void readmatrix (istream & is, Tmatrix & m);	// read a matrix

Trow operator * (double a, Trow r);	// scalar times vector

Tmatrix operator * (double a, Tmatrix m);	// scalar times matrix

Trow operator + (Trow r, const Trow & rr);	// vector + vector

Tmatrix operator + (Tmatrix mm, Tmatrix m);	// matrix + matrix

Trow operator * (const Tmatrix & m, const Trow & r);	// matrix times vector

Trow operator * (const Trow & r, double a);	// vector times scalar

Trow operator / (const Trow & r, double a);	// vector divided by scalar

double dot (const Trow & r, const Trow & rr);

double norm (Trow r);

Tmatrix idmatrix (int n);