Understanding Elementary Row Operations in Linear Systems
This document explains key concepts in linear algebra focusing on elementary row operations used to manipulate systems of equations. It outlines the principles that allow for interchanging the order of equations, multiplying equations by non-zero numbers, and adding or subtracting equations. Through systematic approaches, we can find solutions for linear systems, ensuring that the transformations maintain equivalent systems. The text provides a clear step-by-step guide to effectively reduce linear systems using these operations.
Understanding Elementary Row Operations in Linear Systems
E N D
Presentation Transcript
Chapter 1 Section 1
Examples: 2 x 2 system 2 x 3 system 3 x 2 system
The order in which any two equations are written may be interchanged System 1: System 2: (x1,x2,x3) is a solution to System 1 if any only if it is a solution to to System 2.
Both sides of an equation may be multiplied by the same non zero real number. System 1: System 2: (x1,x2,x3) is a solution to System 1 if any only if it is a solution to to System 2.
A multiple of one equation may be added to (or subtracted from) another. System 1: System 2: *Multiply Equation 2 by 2 and add it to Equation 1 (x1,x2,x3) is a solution to System 1 if any only if it is a solution to to System 2.
The following three operations may be used on a system to obtain an equivalent system: • The order in which any two equations are written may be interchanged. • Both sides of an equation may be multiplied by the same non-zero number • A multiple of one equation may be added to (or subtracted from) an other equation
Elementary Row Operations: • Interchange two rows • Multiply a row by a non-zero real number • Replace a row by its sum with a multiple of an other row
Step 1 Steps for reducing linear systems: Step 2 Step 3
