Ecuaciones

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 todetermine 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

aij xj = bi ,
j=1

i = 1, 2, · · · , m.The matrix

    

a11 a21 . . .

a12 a22

··· ···

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 or not a family of lines (or planes) has a common point of intersection. EXAMPLE 1.1.1 Solve the equation 2x + 3y = 6. Solution. The equation 2x + 3y = 6 is equivalent to 2x = 6 − 3y or 3 x = 3 − 2 y, where y is arbitrary. So there are infinitely many solutions. EXAMPLE 1.1.2 Solve the system x+y+z = 1 x − y + z = 0. Solution. We subtract the secondequation from the first, to get 2y = 1 and y = 1 . Then x = y − z = 1 − z, where z is arbitrary. Again there are 2 2 infinitely many solutions. EXAMPLE 1.1.3 Find a polynomial of the form y = a0 +a1 x+a2 x2 +a3 x3 which passes through the points (−3, −2), (−1, 2), (1, 5), (2, 1).

1.1. INTRODUCTION TO LINEAR EQUATIONS

3

Solution. When x has the values −3, −1, 1, 2, then y takes correspondingvalues −2, 2, 5, 1 and we get four equations in the unknowns a0 , a1 , a2 , a3 : a0 − 3a1 + 9a2 − 27a3 = −2 a0 − a1 + a2 − a3 = 2 a0 + a1 + a2 + a3 = 5 a0 + 2a1 + 4a2 + 8a3 = 1. This system has the unique solution a0 = 93/20, a1 = 221/120, a2 = −23/20, a3 = −41/120. So the required polynomial is y = 93 221 23 41 3 + x − x2 − x . 20 120 20 120

In [26, pages 33–35] there are examples of systems oflinear equations which arise from simple electrical networks using Kirchhoff’s laws for electrical circuits. Solving a system consisting of a single linear equation is easy. However if we are dealing with two or more equations, it is desirable to have a systematic method of determining if the system is consistent and to find all solutions. Instead of restricting ourselves to linear equations withrational or real coefficients, our theory goes over to the more general case where the coefficients belong to an arbitrary field. A field F is a set F which possesses operations of addition and multiplication which satisfy the familiar rules of rational arithmetic. There are ten basic properties that a field must have: THE FIELD AXIOMS. 1. (a + b) + c = a + (b + c) for all a, b, c in F ; 2. (ab)c = a(bc)for all a, b, c in F ; 3. a + b = b + a for all a, b in F ; 4. ab = ba for all a, b in F ; 5. there exists an element 0 in F such that 0 + a = a for all a in F ; 6. there exists an element 1 in F such that 1a = a for all a in F ;

4

CHAPTER 1. LINEAR EQUATIONS 7. to every a in F , there corresponds an additive inverse −a in F , satisfying a + (−a) = 0; 8. to every non–zero a in F , there...