(4.2.1)--2.Eliminationmethodforsolvingali.ppt

Linear AlgebraElimination Method Elimination MethodElimination Method for Solving A Linear System for Solving A Linear SystemThe set of all possible solutions is called the solution set of a linear system.Two linear systems are called equivalent if they have the same solution set.Elimination Method is one of the important methods for solving linear system,applying equivalent transformations to reduce the number of unknows so that the linear system becomes easier to solve.Equivalent transformation doesnt change the solution set of linear system.(1)Interchange any two equations in the system.(2)Multiply any equation by a nonzero scalar.(3)Add a constant multiple of any equation to another.There are three types of equivalent transformations for a system of linear equations:Obviously,the above equivalent transformations dont change the solution set of the linear system.For simplicity,lets see an example with three equations in three unknows.Example Solve the linear system with elimination method.Solution Without loss of generality,the first equation is always denoted as E1,similarly the ith equation is denoted as Ei.E1E2E3During the elimination process,we write out the augmented matrix of each modified system.E1E31/2E2E1E2E3E2-2E1E3-3E1E1E2E3augmented matrixE1E2E3r1r31/2r2r2-2r1r3-3r1E3-2E2E1-E2E1E2E3r3-2r2r1-r2(the reduced row echelon form)Take x3 as a free varibale,then we can obtain the general solution to the original linear system and its vector form.