Vectors, Matrices and Geometry

CHAPTER SEVEN: LINEAR EQUATIONS

CHAPTER SEVEN LINEAR EQUATIONS In this final chapter, we shall apply results of the last two chapters to investigate systems oflinear equations in several unknowns. A necessary and sufficient condition will be given in terms of the ranks of certain matrices and a general method of solution is described. Readers will find that this method is essentially the classical successive eliminations of unknowns but given in terms of elementary row transformations. 7.1 Terminology A linear equation in n unknowns Xl, X2, ... , Xn is an expression of the form (1) where the coefficients a1, a2, ... , an and the constant term b are all fixed real numbers such that at least one coefficient should be non-zero. An ordered n-tuple (t1, t2, .. , ,tn) of real numbers is called a solution of the equation (1) if after substituting each ti for Xi, we get In this case the n-tuple (t1,t2,'" ,tn) or the vector t is said to satisfy the equation (1) and we also say that Xl = t 1, X2 = t2,'" ,Xn = tn constitute a solution to (1). Alternatively if we denote by tERn the vector [t1,t 2, .. · ,tn] and by a E Rn the vector [a1,a2,'" ,an], then t is a solution of the equation (1) if and only if the dot product 281 Vectors, Matrices and Geometry For example IS a linear equation in the 4 unknowns Xl, X2, X3, X4' The 4-tuples (1,1, -1, -1) and (0,1,0, -2) are both solutions of the equations while (0,0,2,0), (1,2, -4, 7) are not. In particular we observe that the equation (1) always has solutions. For example if al :I 0, then Xl = b/al, X2 =X3 =... =Xn =0 constitute a solution to (1). Besides considering one linear equation at a time, we also study simultaneously several linear equations (2) and their common solutions. These rn linear equations constitute a system of rn linear equations in n unknowns Xl, X2, ... , Xn: a11 x l + al2 x 2 + ... + alnxn =bl a2l x l + a22 x 2 + ... + a2n x n = b2 (3) An ordered n-tuple (tl ,t2 ,··· ,tn) is called a solution of the system (3) if after substituting each ti for Xi, we get a11 tl + a12 t 2 + ... + alntn = bl a2l tl + a22 t2 + ... + a2n tn =b2 Alternatively if ai = [ail, ai2, ... , ain] for i = 1,2"" , rn, then t = [tb t2, ... , tn] is a solution of the system (2) if and only if ai . t = bi for i =1, 2, ... , rn. Therefore a solution to the system (3) is a common solution to all equations of the system and vice versa. Our chief concern with a given system of linear equations is clearly that we would like to know if solutions exist, and if they do exist then we would like to have an effective method of evaluating them and a clear and precise way to present them. 282 Linear Equations EXERCISES 1. Show that (1,0, -1, 1), (-1, 1, -1, 1), and (-9,5, -1,1) are all solutions of the system of linear equations 3XI + 6X2 - X3 + X4 = 5 -2XI - 4X2 + X3 = -3 X3 + X4 = o. 2. Verify each of the following: (a) (- ~t - 4, ~t - 6, t) is a solution of the system of linear equations Xl - X2 + X3 = 2 3Xl - X2 + 2X3 = -6 3XI + X2 + X3 = -18 for any real number t. (b) (-s - t, s, -t, 0, t) is a solution of the system of linear equations + Xs = 0 -Xl - X2 + 2X3 - 3X4+ Xs = 0 Xl + X2 - 2X3 - Xs = 0 for any real numbers sand t. 3. Given that (Sl, tl, uI) and (S2, t2, U2) are two solutions of the system of linear equations alxl + a2x2 + a3x3 = 0 blXl + b2X2 + b3X3 = 0 C1Xl + C2X2 + C3X3 = 0 , show that for any real numbers m and n, (msl +nS2, mtl +nt2 , mUl + nU2) is a solution of the system. 7.2 Condition for consistency While every single linear equation always admits solutions, some systems of linear equatoins do not. For example the system Xl + X2 = 1 Xl + X2 = 2 283 Vectors, Matrices and Geometry certainly has no solution. In general we say that a system of linear equations is consistent or solvable if it admits a solution. This leads us to the first important problem of this chapter. That is to find a necessary and sufficient condition for a...