This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: SYSTEMS OF LINEAR EQUATIONS - PART 2 JOS E MALAG ON-L OPEZ We saw that if the coefficient matrix associated to a system of linear equations (S) is in RREF, then it is not difficult to obtain the solution set of (S). In this lecture we will complete the program of finding the solution set for a given system of linear equations. We will develop a method that will produce a new system with the same solutions as the original system, but whose coefficient matrix is in RREF. Such method is called Gaussian elimination. Elementary Row Operations The idea is to consider operations for rows in a matrix such that after performing them we can eliminate entries. If the matrix under consid- eration is the augmented matrix for a system of linear equations, then the elimination of entries will means that we are eliminating variables in the equations. Let A be an m n matrix. The following operations on the rows of A are called the elementary row operations : (1) Interchange any two rows of A . (2) Multiply any row of A by a non-zero number. (3) Add the product of any row A by a number to another row of A . Remark. If A is the augmented matrix associated to a system of linear equations (S), then the operations described above corresponds to: (1) Interchange any two equations in (S). (2) Multiply any equation in (S) by a non-zero number. (3) Add the product of any equation in (S) by a number to another equation in (S). 1 Since it is more convenient to work exclusively with the coefficients, we will describe the method in terms of matrices. Remark. We could have defined the same operations for columns, obtaining similar results. We will restrict ourselves to the case of row operations only. Let A and B be two matrices. We say that A and B are equivalent , and denoted by A B , if we can transform one of the matrices into the other after performing a finite sequence of elementary row operations. Example. We have that parenleftbigg 2 1 6 1 1 1 parenrightbigg parenleftbigg 1 0 7 / 3 0 1 4 / 3 parenrightbigg Indeed, starting with parenleftbigg 2 1 6 1 1 1 parenrightbigg , interchange the two rows: parenleftbigg 2 1 6 1 1 1 parenrightbigg parenleftbigg 1 1 1 2 1 6 parenrightbigg Add -2 times the first row to the second row: parenleftbigg 1 1 1 2 1 6 parenrightbigg parenleftbigg 1 1 1 3 4 parenrightbigg Multiply by 1/3 the second row: parenleftbigg 1 1 1 3 4 parenrightbigg parenleftbigg 1 1 1 1 4 / 3 parenrightbigg Notice that the matrix obtained is in REF. Finally, add the second row to the first row: parenleftbigg 1 1 1 1 4 / 3 parenrightbigg parenleftbigg 1 0 7 / 3 0 1 4 / 3 parenrightbigg...
View Full Document