137(i)Prove thatdet(V)=Yi<j(uj−ui).Conclude thatVis nonsingular if all theuiare distinct.Solution.The proof is by induction onn≥1, but we merelydescribe the inductive step fromn=3ton=4. LetV=1aa2a31bb2b31cc2c31dd2d3.Subtract the top row from each of the other rows (which does notchange the determinant):det(V)=det12a30b−ab2−a2b3−a30c−ac2−a2c3−a30d−ad2−a2d3−a3.By Laplace expansion across the top row, we havedet(V)=detb−2−a2b3−a3c−2−a2c3−a3d−2−a2d3−a3.Use the identitiesx2−y2=(x−y)(x+y)andx3−y3=(x−y)(x2+xy+y2), and factor outb−a,c−a, and
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