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Elementary Linear Algebra Howard Anton & Chris Rorres 1 Chapter Contents 1.1 Introduction to System of Linear Equations 1.2 Gaussian Elimination 1.3 Matrices and Matrix Operations 1.4 Inverses; Rules of Matrix Arithmetic 1.5 Elementary Matrices and a Method for Finding A1 1.6 Further Results on Systems of Equations and Invertibility 1.7 Diagonal, Triangular, and Symmetric Matrices 2 1.1 Introduction to Systems of Equations 3 Linear Equations Any straight line in xy-plane can be represented algebraically by an equation of the form: a1 x a2 y b General form: define a linear equation in the n variables x1 , x2 ,..., xn : a1x1 a2 x2 ... an xn b Where a1 , a2 ,..., an , and b are real constants. The variables in a linear equation are sometimes called unknowns. 4 Example 1 Linear Equations 1 The equations x 3 y 7, y x 3 z 1, and x1 2 x2 3x3 x4 7 are linear. 2 Observe that a linear equation does not involve any products or roots of variables. All variables occur only to the first power and do not appear as arguments for trigonometric, logarithmic, or exponential functions. The equations x 3 y 5, 3x 2 y z xz 4, and y sin x are not linear. A solution of a linear equation is a sequence of n numbers s1 , s2 ,..., sn such that the equation is satisfied. The set of all solutions of the equation is called its solution set or general solution of the equation 5 Example 2 Finding a Solution Set (1/2) Find the solution of (a ) 4 x 2 y 1 Solution(a) we can assign an arbitrary value to x and solve for y , or choose an arbitrary value for y and solve for x .If we follow the first approach and assign x an arbitrary 1 1 value ,we obtain x t , y 2t 1 or x t , y t2 1 1 2 2 2 arbitrary numbers t1, t 2 are called parameter. for example 11 t1 3 yields the solution x 3, y 2 4 as t 2 11 2 6 Example 2 Finding a Solution Set (2/2) Find the solution of (b) x1 4 x2 7 x3 5. Solution(b) we can assign arbitrary values to any two variables and solve for the third variable. for example x1 5 4s 7t , x2 s, x3 t where s, t are arbitrary values 7 Linear Systems (1/2) A finite set of linear equations in the variables x1 , x2 ,..., xn a11x1 a12 x2 ... a1n xn b1 is called a system of linear a21x1 a22 x2 ... a2 n xn b2 equations or a linear system . A sequence of numbers s1 , s2 ,..., sn is called a solution of the system. A system has no solution is said to be inconsistent ; if there is at least one solution of the system, it is called consistent. am1 x1 am 2 x2 ... amn xn bm An arbitrary system of m linear equations in n unknowns 8 Linear Systems (2/2) Every system of linear equations has either no solutions, exactly one solution, or infinitely many solutions. A general system of two linear equations: (Figure1.1.1) a1 x b1 y c1 (a1 , b1 not both zero) a2 x b2 y c2 (a2 , b2 not both zero) Two lines may be parallel -> no solution Two lines may intersect at only one point -> one solution Two lines may coincide -> infinitely many solution 9 Augmented Matrices The location of the +’s, the x’s, and the =‘s can be abbreviated by writing only the rectangular array of numbers. This is called the augmented matrix for the system. Note: must be written in the same order in each equation as the unknowns and the constants must be on the right. a11x1 a12 x2 ... a1n xn b1 a21 x1 a22 x2 ... a2 n xn b2 am1 x1 am 2 x2 ... amn xn bm 1th column a11 a12 ... a1n b1 a a ... a b 2n 2 21 22 a a ... a b mn m m1 m 2 1th row 10 Elementary Row Operations The basic method for solving a system of linear equations is to replace the given system by a new system that has the same solution set but which is easier to solve. Since the rows of an augmented matrix correspond to the equations in the associated system. new systems is generally obtained in a series of steps by applying the following three types of operations to eliminate unknowns systematically. These are called elementary row operations. 1. Multiply an equation through by an nonzero constant. 2. Interchange two equation. 3. Add a multiple of one equation to another. 11 Example 3 Using Elementary row Operations(1/4) x y 2z 9 2 x 4 y 3z 1 3x 6 y 5 z 0 1 1 2 9 2 4 3 1 3 6 5 0 add - 2 times the first equation to the second x y 2z 9 2 y 7 z 1 7 3x 6 y 5 z add - 2 times the first row to the second 0 add -3 times the first equation to the third 9 1 1 2 0 2 7 17 3 6 5 0 add -3 times the first row to the third 12 Example 3 Using Elementary row Operations(2/4) add -3 times x y 2 z 9 multiply the second x y 2 z 9 1 the second equation equation by 7 17 2 y 7 z 17 y 2 z 2 to the third 2 3 y 11z 27 3 y 11z 0 multily the second 9 1 1 2 1 0 2 7 17 row by 2 0 3 11 27 9 add -3 times 1 1 2 0 1 7 17 the second row 2 2 to the third 0 3 11 27 13 Example 3 Using Elementary row Operations(3/4) x y 2z 9 Multiply the third equation by -2 x y 2z 9 7 17 y z y 72 z 172 2 2 z 3 12 z 32 1 1 2 0 1 7 2 0 0 12 9 172 32 Multily the third row by -2 1 1 2 0 1 7 2 0 0 1 Add -1 times the second equation to the first 9 Add -1 times the 172 second row to the first 3 14 Example 3 Using Elementary row Operations(4/4) x 112 z 35 2 y 72 z 172 z 1 0 112 0 1 7 2 0 0 1 3 3 35 2 17 2 Add - 121 times the third equation to the first and 72 times the third equation to the second Add - 121 times the third row to the first and 72 times the third row to the second x y 1 2 z 3 1 0 0 1 0 1 0 2 0 0 1 3 The solution x=1,y=2,z=3 is now evident. 15 Exercise Set 1.1 Question 3 Add - 121 times the third equation to the first and 72 times the third equation to the second 16 Exercise Set 1.1 Question 8 17 Exercise Set 1.1 Question 11 18