Adv Alegbra HW Solutions 138

Adv Alegbra HW Solutions 138 - 138(ii If is a primitive n...

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138 (ii) If ω is a primitive n th root of unity ( ω n = 1 and ω i ±= 1for i < n ), prove that Van ( 1 ,ω,ω 2 ,...,ω n 1 ) is nonsingular and that Van ( 1 2 n 1 ) 1 = 1 n ( 1 1 2 n + 1 ). Solution. Absent. (iii) Let f ( x ) = a 0 + a 1 x + a 2 x 2 +···+ a n x n k [ x ] , and let y i = f ( u i ) . Prove that the coef f cient vector a = ( a 0 ,..., a n ) is a solution of the linear system Vx = y ,( 4 ) where y = ( y 0 y n ) . Conclude that if all the u i are distinct, then f ( x ) is determined by Eq. (4). Solution. Absent. 4.53 Let T [ x 1 x n ] be an n × n tridiagonal matrix. (i) If D n = det ( T [ x 1 x n ] ) , prove that D 1 = x 1 , D 2 = x 1 x 2 + 1, and, for all n > 2, D n = x n D n 1 + D n 2 . Solution. Absent. (ii) Prove that if all x i = 1, then D n = F n + 1 , the n th Fibonacci num- ber. Solution. Absent. 4.54 Let T : C n C n be a linear transformation, and let A =[ a ij ] be its matrix relative to the standard basis e 1 e n . (i)
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