Adv Alegbra HW Solutions 137

Adv Alegbra HW Solutions 137 - 137 (i) Prove that (u j u i...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
137 (i) Prove that det ( V ) = Y i < j ( u j u i ). Conclude that V is nonsingular if all the u i are distinct. Solution. The proof is by induction on n 1, but we merely describe the inductive step from n = 3to n = 4. Let V = 1 aa 2 a 3 1 bb 2 b 3 1 cc 2 c 3 1 dd 2 d 3 . Subtract the top row from each of the other rows (which does not change the determinant): det ( V ) = det 1 2 a 3 0 b ab 2 a 2 b 3 a 3 0 c ac 2 a 2 c 3 a 3 0 d ad 2 a 2 d 3 a 3 . By Laplace expansion across the top row, we have det ( V ) = det b 2 a 2 b 3 a 3 c 2 a 2 c 3 a 3 d 2 a 2 d 3 a 3 . Use the identities x 2 y 2 = ( x y )( x + y ) and x 3 y 3 = ( x y )( x 2 + xy + y 2 ) , and factor out b a , c a , and
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.
Ask a homework question - tutors are online