Chapter 1.1 SystemsLinearEquations

Chapter 1.1 SystemsLinearEquations - Outline Linear Systems...

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Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Solving Linear Systems of Equations Math 2270 K. J. Platt, Ph.D. Department of Mathematics Snow College Spring 2014

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Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Outline 1 Linear Systems Linear Equations Linear Systems of Equations 2 Matrices 3 Gaussian Elimination Echelon Forms 4 Homogeneous Systems
Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Class Expectations Regular Class Attendance. Be Familiar With the Course Syllabus Ask Questions. Take Good Notes. Be Respectful. Spend Adequate Time Outside of Class On Learning. Complete All Assignments and Exams On Time. Come To My Office Hours for Additional Help. HAVE FUN LEARNING MATH!

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Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Linear Equations Definition A linear equation in n variables is an equation of the form a 1 x 1 + a 2 x 2 + · · · + a n x n = b , where a 1 , a 2 , . . . , a n , b are fixed real numbers, and x 1 , x 2 , . . . , x n are variables, called unknowns . Example Which equations are linear? 1 2 x - 3 y = 7 2 - 5 x + 2 y - 7 z = 20 3 x + 4 y 2 = 9 4 ln x + sin y = - 2
Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Solving Linear Equations Consider a linear equation a 1 x 1 + a 2 x 2 + · · · + a n x n = b . A solution of the equation is an ordered list of numbers s 1 , s 2 , . . . , s n such that if x 1 = s 1 , x 2 = s 2 , . . . , x n = s n , then the equation yields a true statement. To solve the equation is to find all solutions of the equation. The set of all solutions of the equation is called the solution set or general solution of the equation. If n 2, the equation has infinitely many solutions. We describe the solutions via parametric equations by assigning special variables, called parameters , to n - 1 of the variables, then expressing the last variable in terms of the parameters.

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Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Examples Example Solve the linear equations. 1 3 x + 6 y = 20 2 x - 4 y + 9 z = 5
Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Linear Systems of Equations A collection of m linear equations in n unknowns is called a system of linear equations . We can write the system as: a 11 x 1 + a 12 x 2 + · · · + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + · · · + a 2 n x n = b 2 . . . . . . . . . . . . a m 1 x 1 + a m 2 x 2 + · · · + a mn x n = b m (1) The real number a ij is the coefficient of x j in the i th equation. The real number b i is the constant term in the i th equation. A solution of the system is an ordered list of numbers s 1 , s 2 , . . . , s n that is a solution of each equation in the system.

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