lecture_slides_1.1-1.3.pdf - Math 220 Robert Creese September 3 2019 Robert Creese September 3 2019 1 62 1.1 Systems of linear equations Robert Creese

lecture_slides_1.1-1.3.pdf - Math 220 Robert Creese...

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Math 220 Robert Creese September 3, 2019 Robert Creese September 3, 2019 1 / 62
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1.1 Systems of linear equations Robert Creese September 3, 2019 2 / 62
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1.1 Linear Equation A Linear Equation in variables x 1 , x 2 , · · · , x n is an equation that can be written as a 1 x 2 + a 2 x 2 + · · · a n x n = b . (1) Here a 1 , a 2 , · · · a n are called coefficients. The coefficients and b are normally known and are real or complex numbers. For example 4 x 1 + 5 x 2 = - 1 + x 2 is a linear equation since it can be written in the form in (1). However x 1 x 2 + 3 x 1 - 4 x 2 = 3 is not a linear equation. Note that we cannot multiply variables with other variables or use exponents. Robert Creese September 3, 2019 3 / 62
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1.1 Systems of linear equations A System of Linear Equations in variables x 1 , x 2 , · · · , x n is a set of one or more linear equations in variables x 1 , x 2 , · · · , x n . An example of a system of linear equations is x 1 +2 x 2 - x 3 = 0 2 x 1 +4 x 3 = 6 (2) Note that the number of variables and number of equations may not necessarily match. Robert Creese September 3, 2019 4 / 62
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1.1 Systems of linear equations A solution of a system , is a set of numbers ( s 1 , s 2 , · · · s n ) where each equation is true if x 1 = s 1 , · · · x n = s n . For example (1 , 0 , 1) is a solution the system x 1 +2 x 2 - x 3 = 0 2 x 1 +4 x 3 = 6 (3) The solution set of a system is the set of all possible solutions to that system. . Two linear systems are called equivalent if they share the same solution set. Robert Creese September 3, 2019 5 / 62
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1.1 Systems of linear equations A system of linear equations will always have either 1 No solution 2 One solution 3 Infinitely many solutions This can be illustrated in the case where have system of two equations with two variables. In this case each equations represents a line and each solution is a point where the lines intersect. So there are three cases 1 Parallel Lines (never intersect so no solution) Ex. x 2 = x 1 , x 2 = 1 + x 1 2 Intersecting lines (Intersect at one point so one solution) Ex. x 2 = x 1 , x 2 = 2 x 1 3 Coinciding lines (lines are the same so every point on the line is a solution, infinite solutions) Ex. x 2 = x 1 , 2 x 2 = 2 x 1 Robert Creese September 3, 2019 6 / 62
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1.1 Systems of linear equations A system of linear equations is consistent if it has at least one solution. A system of linear equations is inconsistent if it has no solution. Systems of linear equations will often be referred to as a system frequently for the purpose of brevity. Robert Creese September 3, 2019 7 / 62
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1.1 Matrix Notation Here a matrix is a rectangular array of numbers. The shape of a matrix is important. If a matrix A has m rows and n columns, then A has size m × n and is a m × n matrix. For example the following linear system x 1 +4 x 2 +3 x 3 = 2 2 x 2 - x 3 = - 10 - 2 x 1 + x 2 = - 7 (4) has a coefficient matrix (matrix including the coefficients of a linear system) 1 4 3 0 2 - 1 - 2 1 0 There is a 0 in the second row, first column since there is no x 1 in the second equation ( x 1 has coefficient 0).
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