如果a1,a2,a3线性无关,证明2a1+3a2,a2+4a3,5a3+a1线性相关

题目

如果a1,a2,a3线性无关,证明2a1+3a2,a2+4a3,5a3+a1线性相关

答案

令0 = x[2a1 + 3a2] + y[a2 + 4a3] + z[a1 + 5a3]= a1[2x + z] + a2[3x + y] + a3[4y + 5z]因 a1,a2,a3线性无关.所以2x + z = 0 = 3x + y = 4y + 5z,z = -2x,y = -3x,0 = 4y + 5z = 4(-3x) + 5(-2x) = -22x,x = 0.y ...