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Lecture 8

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MATH 136
Robert Sproule

Wednesday, January 22 − Lecture 8 : Systems of linear equations. Concepts: 1. m linear equations in n unknowns (system of linear equations). 2. solving a system of linear equations 3. equivalent system of linear equations 4. coefficient matrix, augmented coefficient matrix. 5. RREF coefficient matrix, pivot, leading-1, REF matrix 8.1 Introduction − Linear equations are equations of the form a1 1+ a 2 2 ... + a xn nb. where the x’siare called the variables, or unknowns, while the a ’s ari the coefficients of the linear equation. The term b which is not juxtaposed to a variable is referred to as the constant.  A system of linear equations is a finite set of linear equations each containing the same variables.  “Solving a system of linear equations” means finding the set of all vector values x = (x1, x2, 3 , …, xn) that satisfy all of the given linear equations.  In this lecture we discuss a method for solving these by manipulating only the coefficients and the constants.  To solve a system of linear equations we apply repeatedly the following fundamental mathematical principles: 8.1.1 Example − In the following example we solve a system of three linear equations in three unknowns by applying these principles. Solve the system 4x − 8y + −4z = 36 2x − y + z = 6 3x − 2y + −2z = 2. Solution : Denote each equation as follows : Each of the systems listed above have the same solution. The last system is the one that is easiest to solve. 8.1.2 Definition – Two systems of linear equations are said to be equivalent systems if they have the same set of solutions. In the example of above we have step by step transformed a system of equations to another. Each time we this transformation did not change the solution set of the system. Each one of the systems of equations which appear above are equivalent systems. 8.1.3 Geometric interpretation.  An equation with three unknowns describes a plane in 3-space.  When three equations in three unknowns are considered simultaneously to form a “system of equations” the solution set to this system describes all points which belong to the three planes. This solution set can be empty, a single point, a line or a plane. Representing a system of linear equations as a rectangular array of numbers. 8.2 Definition − System of linear equations.
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