Linear dependence and independence VI Since a;=(x1i, X2i, ., Xni),then c1a1+c2a2+…+cmm=0. It equal to the system of equations C1x1+c2×2+……+cmX1m=0 C1x21+c2x2+…+cm×m=0 c1xn+c2Xn+…+cmXm=0 Let x=(c1, C2,..., Cm), then the matrix equation is Ax=0 From the theorem, a linear equation Ax=0 has nontrivial if and only if rca<mLinear dependence and independence VI Proof. Since αi = (x1i , x2i , · · · , xni) T , then c1α1 + c2α2 + · · · + cmαm = 0. It equal to the system of equations    c1x11 + c2x12 + · · · + cmx1m = 0 c1x21 + c2x22 + · · · + cmx2m = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c1xn1 + c2xn2 + · · · + cmxnm = 0 . Let x = (c1, c2, · · · , cm), then the matrix equation is Ax = 0. From the theorem, a linear equation Ax = 0 has nontrivial if and only if R(A) < m. () May 3, 2006 6 / 40