The Structure of solutions First we look at a homogeneous system a111+a12X+…+a1nXn=0 a21X1+a22X+…+a2nxn=0 am1x1 t am2X2 +..+ amn=0
The Structure of Solutions First we look at a homogeneous system a11x1 + a12x2 + · · · + a1nxn = 0 a21x1 + a22x2 + · · · + a2nxn = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . am1x1 + am2x2 + · · · + amnxn = 0 (2) The set of solutions of a system of linear equations is called the solution set. The solution set of a homogenous system is also called the solution space. If it has zero solution only, the structure is simple and we are not interested in this case. So we suppose that the rank r of the coefficient matrix of (2) is less than n. Then the system has infinitely many solutions. () April 28, 2006 1 / 15
The Structure of solutions First we look at a homogeneous system a111+a12X+…+a1nXn=0 a21X1+a22X+…+a2nxn=0 am1X1 t am2X2+..+ amnon=0 o The set of solutions of a system of linear equations is called the solution set
The Structure of Solutions First we look at a homogeneous system a11x1 + a12x2 + · · · + a1nxn = 0 a21x1 + a22x2 + · · · + a2nxn = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . am1x1 + am2x2 + · · · + amnxn = 0 (2) The set of solutions of a system of linear equations is called the solution set. The solution set of a homogenous system is also called the solution space. If it has zero solution only, the structure is simple and we are not interested in this case. So we suppose that the rank r of the coefficient matrix of (2) is less than n. Then the system has infinitely many solutions. () April 28, 2006 1 / 15
The Structure of solutions First we look at a homogeneous system a111+a12X+…+a1nXn=0 a21X1+a22X+…+a2nxn=0 am1X1 t am2X2+..+ amnon=0 o The set of solutions of a system of linear equations is called the solution set o The solution set of a homogenous system is also called the solution space
The Structure of Solutions First we look at a homogeneous system a11x1 + a12x2 + · · · + a1nxn = 0 a21x1 + a22x2 + · · · + a2nxn = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . am1x1 + am2x2 + · · · + amnxn = 0 (2) The set of solutions of a system of linear equations is called the solution set. The solution set of a homogenous system is also called the solution space. If it has zero solution only, the structure is simple and we are not interested in this case. So we suppose that the rank r of the coefficient matrix of (2) is less than n. Then the system has infinitely many solutions. () April 28, 2006 1 / 15
The Structure of solutions First we look at a homogeneous system a111+a12X+…+a1nXn=0 a21X1+a22X+…+a2nxn=0 am1X1 t am2X2+..+ amnon=0 o The set of solutions of a system of linear equations is called the solution set o The solution set of a homogenous system is also called the solution space o If it has zero solution only, the structure is simple and we are not interested in this case
The Structure of Solutions First we look at a homogeneous system a11x1 + a12x2 + · · · + a1nxn = 0 a21x1 + a22x2 + · · · + a2nxn = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . am1x1 + am2x2 + · · · + amnxn = 0 (2) The set of solutions of a system of linear equations is called the solution set. The solution set of a homogenous system is also called the solution space. If it has zero solution only, the structure is simple and we are not interested in this case. So we suppose that the rank r of the coefficient matrix of (2) is less than n. Then the system has infinitely many solutions. () April 28, 2006 1 / 15
The Structure of solutions First we look at a homogeneous system a111+a12X+…+a1nXn=0 a21X1+a22X+…+a2nxn=0 am1X1 t am2X2+..+ amnon=0 o The set of solutions of a system of linear equations is called the solution set o The solution set of a homogenous system is also called the solution space o If it has zero solution only, the structure is simple and we are not interested in this case o So we suppose that the rank r of the coefficient matrix of(2 )is less than n Then the system has infinitely many solutions
The Structure of Solutions First we look at a homogeneous system a11x1 + a12x2 + · · · + a1nxn = 0 a21x1 + a22x2 + · · · + a2nxn = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . am1x1 + am2x2 + · · · + amnxn = 0 (2) The set of solutions of a system of linear equations is called the solution set. The solution set of a homogenous system is also called the solution space. If it has zero solution only, the structure is simple and we are not interested in this case. So we suppose that the rank r of the coefficient matrix of (2) is less than n. Then the system has infinitely many solutions. () April 28, 2006 1 / 15