1 - Systems of Linear Equations
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1 1 - Systems of Linear Equations 1.1 Introduction to Systems of Linear Equations Almost every problem in linear algebra will involve solving a system of equations. ü LINEAR EQUATIONS IN n VARIABLES We are used to dealing with linear equations in two variables x and y, such as a 1 x + a 2 y = b, and perhaps even linear equations in three variables x, y, and z such as a 1 x + a 2 y + a 3 z = b. More generally Definition of a Linear Equation in n Variables A linear equation in n variables x 1, x 2,..., x n has the form a 1 x 1 + a 2 x a n x n = b. The coefficients a 1, a 2,..., a n are real numbers, and the constant term b is a real number. The number a 1 is the leading coefficient, and x 1 is the leading variable. ü SOLUTIONS AND SOLUTION SETS A solution of a linear equation in n variables is a sequence of n real numbers s 1, s 2,..., s n, such that when substituted for the variables x 1, x 2,..., x n respectively, in the linear equation, they will satisfy the equation. A solution set is the set of all solutions to an equation. ü SYSTEMS OF LINEAR EQUATIONS A system of m linear equations in n variables can be expressed in the form a 11 x 1 + a 12 x 2 + a 13 x a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + a 23 x a 2 n x n = b 2 a 31 x 1 + a 32 x 2 + a 33 x a 3 n x n = b 3 ª a m 1 x 1 + a m 2 x 2 + a m 3 x a mn x n = b m A solution of a system of m linear equation in n variables is a sequence of n real numbers s 1, s 2,..., s n that is a solution of each equation. y y y x x x One Solution Infinite Solutions No Solution
2 2 MATH_2318_CH_01.nb Number of Solutions of a Linear Equations For a system of linear equations, precisely one of the following is true. 1. The system has exactly one solution (consistent system). 2. The system has infinitely many solutions (consistent system). 3. The system has no solution (inconsistent system). In the case of one solution, we can represent this solution by an ordered n-tuple, Hs 1, s 2,..., s n L. In the case of infinite solutions, we use a parametric representation to define our solution set. ü Example Find a parametric representation of the solution set of the linear equation x - 3 y + 2 z = 1. ü SOLVING A SYSTEM OF LINEAR EQUATIONS Recall that you learned two algebraic techniques: substitution and addition/elimination. Of the two, addition/elimination is more useful. Operations That Produce Equivalent Systems Each of the following operations on a system of linear equations produces an equivalent system. 1. Interchange two equations. 2. Multiply an equation by a nonzero constant. 3. Add a multiple of an equation to another equation. ü Example Use the addition/elimination method to solve the following system of equations. 3 x + 2 y = 5 x - 2 y = 3 While good for solving 2 equations in 2 variables and adequate for solving 3 equations in 3 variables, this algebraic method becomes cumbersome for larger systems. Matrices will make this method much easier.
3 MATH_2318_CH_01.nb Gaussian Elimination and Gauss-Jordan Elimination ü MATRICES Definition of a Matrix If m and n are positive integers, an mμn matrix is a rectangular array a 11 a 12 a 13 a 1 n a 21 a 22 a 23 a 2 n A = a 31 a 32 a 33 a 3 n ª ª ª ª a m 1 a m 2 a m 3 a mn in which each entry, a ij, of the matrix is a number. An mμn matrix has m rows and n columns. Matrices are usually denoted by capital letters. The size (or dimension) of a matrix is given by mμn. If m = n then we often refer to the matrix as a square matrix of order n. In a square matrix, the entries a 11, a 22,..., a nn are called the main diagonal entries. A system of m equations in n variables can be represented using a matrix. Given the system the coefficient matrix is given by a 11 x 1 + a 12 x 2 + a 13 x a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + a 23 x a 2 n x n = b 2 a 31 x 1 + a 32 x 2 + a 33 x a 3 n x n = b 3 ª a m 1 x 1 + a m 2 x 2 + a m 3 x a mn x n = b m The augmented matrix for the system is given by a 11 a 12 a 13 a 1 n a 21 a 22 a 23 a 2 n a 31 a 32 a 33 a 3 n ª ª ª ª a m 1 a m 2 a m 3 a mn a 11 a 12 a 13 a 1 n b 1 a 21 a 22 a 23 a 2 n b 2 a 31 a 32 a 33 a 3 n b 3 ª ª ª ª ª a m 1 a m 2 a m 3 a mn b m
4 4 MATH_2318_CH_01.nb ü Example Give the coefficient and augmented matrices for the system of equations. 2 x 2 - x 3 = 14 2 x 1 - x x 3 = 24 7 x 1-5 x 2 = 6 ü ELEMENTARY ROW OPERATIONS Since a row represents an equation in augmented matrix, we can restate the operations that produce equivalent systems as row operations that produce a matrix that is row-equivalent. Two matrices are considered to be row-equivalent if one can be obtained from the other by an elementary row operation. Elementary Row Operations 1. Interchange two rows. 2. Multiply a row by a nonzero constant. 3. Add a multiple of a row to another row. ü Example Perform the indicated row operations on the matrix a. Interchange row 1 and row 2. b. With the result of part a., multiply row 1 by 1 2. c. With the result of part b., add -7 times row 1 to row 3. Our goal is to obtain a matrix that is row-equivalent to the original matrix but one which will easily give us the solution(s), if they exist.
5 MATH_2318_CH_01.nb 5 Row-Echelon and Reduced Row-Echelon Form A matrix in row-echelon form has the following properties. 1. Any rows consisting entirely of zeros occur at the bottom of the matrix. 2. For each row that does not consist entirely of zeros, the first nonzero entry is 1 (called a leading 1). 3. For two successive (nonzero) rows, the leading 1 in the higher row is farther to the left than the leading 1 in the lower row. A matrix in row-echelon form is in reduced row-echelon form when every column that has a leading 1 has zeros in every position above and below its leading 1. ü Example Perform the necessary row operations to the matrix from part c of Example to obtain a row-equivalent matrix in row-echelon form. To find the solution, we need to use back-substitution. ü Example Use back-substitution with the result from Example to find the solution to the system of equations. This process of solving for solving a system of equations is known as Gaussian Elimination with Back-Substitution. Gaussian Elimination with Back-Substitution 1. Write the augmented matrix of the system of linear equations. 2. Use elementary row operations to rewrite the matrix in row-echelon form. 3. Write the system of linear equations corresponding to the matrix in row-echelon form, and use back-substitution to find the solution.
6 6 MATH_2318_CH_01.nb ü GAUSS-JORDAN ELIMINATION A similar process of solving for solving a system of equations, known as Gauss-Jordan Elimination, requires that you obtain the reduced row-echelon form of the augmented matrix. ü Example Use Gauss-Jordan elimination to solve the system of equations. 3 x 1-2 x x 3 = 10 x 1 + x 2-2 x 3 = 3 2 x 1-3 x x 3 = 8 ü HOMOGENEOUS SYSTEMS OF LINEAR EQUATIONS A system of m linear equations in n variables of the form a 11 x 1 + a 12 x 2 + a 13 x a 1 n x n = 0 a 21 x 1 + a 22 x 2 + a 23 x a 2 n x n = 0 a 31 x 1 + a 32 x 2 + a 33 x a 3 n x n = 0 ª a m 1 x 1 + a m 2 x 2 + a m 3 x a mn x n = 0 is called a homogeneous system of equations. Note that it always has at least one solution, which is the trivial solution (when the value of all variables is zero).
7 MATH_2318_CH_01.nb 7 Theorem 1.1 The Number of Solutions of a Homogeneous System Every homogeneous system of linear equations is consistent. Moreover, if the system has fewer equations than variables, it must have infinitely many solutions. ü Example Solve the system of equations. 5 x x 2 - x 3 = 0 10 x x x 3 = 0 5 x x 2-9 x 3 = 0
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