Advanced Calculus fi..

14 Advanced Calculus, Fifth EditionIf D = 0, then (1.39) and (1.40) show that, if there is a solution, then Dl = 0,D2 = 0, 0 3 = 0. We reserve discussion of the general case D = 0 to Section 1.10and here consider only the homogeneous systema31x + a32y + a3sz = 0.Here DI = 0, D2 = 0, D3 = 0. Thus if D # 0, Eqs. (1.41) have the unique solutioncalled the trivial solution.On the other hand, if D = 0, then Eqs. (1.41) have infinitely many solutions.To show this, we introduce the vectorsThenxv, +yv2+zv3 hascomponentsa~~x+a~~y +al32, azlx+a22y +a2329 a3,x+a3~y + ~332. Hence Eqs. (1.41) are equivalent to the vector equationNow we have assumed that D = 0. It follows that vl, v2, v3 are linearly dependent.For the corresponding determinant,v1 ' v2 x v3equals D with rows and columns interchanged; by Rule I of Section 1.4, this determinantequals D and thus is 0. Therefore v,, v2, v3 are linearly dependent, andnumbers cl, c2, c3 that are not all 0 can be found such thatThus .x = CII, y = c2t, z = c3t, where t is arbitrary, provides infinitely manysolutions of (1.43) and hence of (1.41).The results established here extend to the general case of n equations in nunknowns:Here