Elementary Linear Algebra

ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND First Printing, 1991 Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x1 , x2 , · · · , xn is an equation of the form a1 x1 + a2 x2 + · · · + an xn = b, where a1 , a2 , . . . , an , b are given real numbers. For example, with x and y instead of x1 and x2 , the linear equation 2x + 3y = 6 describes the line passing through the points (3, 0) and (0, 2). Similarly, with x, y and z instead of x1 , x2 and x3 , the linear equation 2x + 3y + 4z = 12 describes the plane passing through the points (6, 0, 0), (0, 4, 0), (0, 0, 3). A system of m linear equations in n unknowns x1 , x2 , · · · , xn is a family of linear equations a11 x1 + a12 x2 + · · · + a1n xn = b1 a21 x1 + a22 x2 + · · · + a2n xn = b2 .. . am1 x1 + am2 x2 + · · · + amn xn = bm . We wish to determine if such a system has a solution, that is to find out if there exist numbers x1 , x2 , · · · , xn which satisfy each of the equations simultaneously. We say that the system is consistent if it has a solution. Otherwise the system is called inconsistent. 1 2 CHAPTER 1. LINEAR EQUATIONS Note that the above system can be written concisely as n X aij xj = bi , j=1 The matrix      a11 a21 .. . a12 a22 i = 1, 2, · · · , m. ··· ··· a1n a2n .. . am1 am2 · · · amn      is called the coefficient matrix of the system, while the matrix   a11 a12 · · · a1n b1  a21 a22 · · · a2n b2     .. .. ..   . . .  am1 am2 · · · amn bm is called the augmented matrix of the system. Geometrically, solving a system of linear equations in two (or three) unknowns is equivalent to determining whether