Lecture 3 Gauss Jordan Method Homogeneous Linear System

Lecture 3 Gauss Jordan Method Homogeneous Linear System

1. 2 Gauss Jordran Method 2

Example 4 Gauss-Jordan Elimination(1/4) • Solve by Gauss-Jordan Elimination • Solution: The augmented matrix for the system is 3

Example 4 Gauss-Jordan Elimination(2/4) • Adding -2 times the 1 st row to the 2 nd and 4 th rows gives • Multiplying the 2 nd row by -1 and then adding -5 times the new 2 nd row to the 3 rd row and -4 times the new 2 nd row to the 4 th row gives 4

Example 4 Gauss-Jordan Elimination(3/4) • Interchanging the 3 rd and 4 th rows and then multiplying the 3 rd row of the resulting matrix by 1/6 gives the row-echelon form. • Adding -3 times the 3 rd row to the 2 nd row and then adding 2 times the 2 nd row of the resulting matrix to the 1 st row yields the reduced rowechelon form. 5

Example 4 Gauss-Jordan Elimination(4/4) • The corresponding system of equations is • Solution The augmented matrix for the system is • We assign the free variables, and the general solution is given by the formulas: 6

1. 2 Homogeneous Linear System 7

Homogeneous Linear Systems(1/2) • A system of linear equations is said to be homogeneous if the constant terms are all zero; that is , the system has the form : • Every homogeneous system of linear equation is consistent, since all such system have as a solution. This solution is called the trivial solution; if there another solutions, they are called nontrivial solutions. • There are only two possibilities for its solutions: • The system has only the trivial solution. • The system has infinitely many solutions in addition to the trivial solution. 8

Homogeneous Linear Systems(2/2) • In a special case of a homogeneous linear system of two linear equations in two unknowns: (fig 1. 2. 1) 9

Example 7 Gauss-Jordan Elimination(1/3) n n Solve the following homogeneous system of linear equations by using Gauss. Jordan elimination. Solution n n The augmented matrix Reducing this matrix to reduced row-echelon form 10

Example 7 Gauss-Jordan Elimination(2/3) Solution (cont) n The corresponding system of equation n Solving for the leading variables is n Thus the general solution is n Note: the trivial solution is obtained when s=t=0. 11

Theorem 1. 2. 1 A homogeneous system of linear equations with more unknowns than equations has infinitely many solutions. • Note: theorem 1. 2. 1 applies only to homogeneous system • Example 7 (3/3) 12

Homogeneous Linear Systems • A system of linear equations is said to be homogeneous if the constant terms are all zero. • Every homogeneous sytem of linear equations is consistent, since all such systems have x 1=0, x 2=0, . . . , xn=0 as a solution [trivial solution]. Other solutions are called nontrivial solutions. Linear Algebra - Chapter 1 13

Homogeneous Linear Systems • Example: [Gauss-Jordan Elimination] Linear Algebra - Chapter 1 14

Homogeneous Linear Systems The corresponding system of equations is Solving for the leading variables yields The general solution is the trivial solution is obtained when s=t=0 Linear Algebra - Chapter 1 15

Homogeneous Linear Systems • Theorem: A homogeneous system of linear equations with more unknowns than equations has infinitely many solutions. Linear Algebra - Chapter 1 16