Sp13_2568_C1_S1and2_handout - 1.1 and1.2 Chapter 1 Matrices and Systems of Linear Equations 1.1 Introduction to Matrices and Systems of Linear Equations

# Sp13_2568_C1_S1and2_handout - 1.1 and1.2 Chapter 1 Matrices...

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§ 1.1 and § 1.2 1.1 Chapter 1 Matrices and Systems of Linear Equations § 1.1: Introduction to Matrices and Systems of Linear Equations § 1.2: Echelon Form and Gauss-Jordan Elimination Lecture Linear Algebra - Math 2568M on Friday, January 11, 2013 Oguz Kurt MW 605 [email protected] 292-9659 Off. Hrs: MWF 10:20-11:20 The Ohio State University
§ 1.1 and § 1.2 1.2 Outline 1 § 1.1 and § 1.2
§ 1.1 and § 1.2 Linear Equations Definition
§ 1.1 and § 1.2 1.4 Systems of Linear Equations (SLEs) Definition A system of linear equations is a finite set of linear equations, each with the same variables. A solution of system of linear equations is a vector that is simultaneously a solution of each equation in the system. The solution set of a system of linear equations is the set of all solutions of the system. Example The system 2 x - 3 y = 7 3 x + y = 5 has [ 2 , - 1 ] as a solution.
§ 1.1 and § 1.2 1.5 Consistency and type of solutions Note: A system of linear equations is called consistent if it has at least one solution. A system with no solutions is called inconsistent . A system of linear equations with real coefficients has either 1 a unique solution (a consistent system) or 2 infinitely many solutions (a consistent system) or 3 no solutions (an inconsistent system).
§ 1.1 and § 1.2 1.6 Solving a System of Linear Equations Example: Solve the system x - y - z = 4 2 y + z = 5 3 z = 9 Note: To solve this system, we usually use back substitution . 3 z = 9 z = 3 z = 3 , 2 y + z = 5 2 y = 2 y = 1 z = 3 , y = 1 , x - y - z = 4 x = 8 [ 8 , 1 , 3 ] is the unique solution for this SLEs.
§1.1 and§1.21.7Using Augmented MatrixAssume we have the following SLEs withmequations andnunknowns:a1,1x1+a1,2x2...a1,nxn=b1a2,1x1+a2,2x2...a2,nxn=b2...am,1x1+an,2x2...am,nxn=bmWe can represent it as follows:x1x2......xnEq. 1a1,1a1,2......a1,nb1Eq. 2a2,1a2,2......a2,nb2...............Eq. mam,1am,2......am,nbm
§ 1.1 and § 1.2 1.8 Augmented Matrix The following matrix is called the augmented matrix of this SLEs: a 1 , 1 a 1 , 2 ... ... a 1 , n b 1 a 2 , 1 a 2 , 2 ... ... a 2 , n b 2 . . . . . . . . . . . . a m , 1 a m , 2 ... ... a m , n b m Augmented Matrix can be used to solve SLEs.