Adv Alegbra HW Solutions 137

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

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137 (i) Prove that det ( V ) = 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 = 3 to n = 4. Let V = 1 a a 2 a 3 1 b b 2 b 3 1 c c 2 c 3 1 d d 2 d 3 . Subtract the top row from each of the other rows (which does not change the determinant): det ( V ) = det 1 a a 2 a 3 0 b a b 2 a 2 b 3 a 3 0 c a c 2 a 2 c 3 a 3 0 d a d 2 a 2 d 3 a 3 . By Laplace expansion across the top row, we have det ( V ) = det b a b 2 a 2 b 3 a 3 c a c 2 a 2 c 3 a 3 d a 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 d
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