Q3. Let a1, 02,..., an be distinct elements of a field K. Consider the Vandermonde matrix...

 

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Q3. Let a1, 02,..., an be distinct elements of a field K. Consider the Vandermonde matrix 1 αι αξ 1 -(45) 1 an an V := (i) Compute the determinant det (V) when n = 2. (ii) Use induction to show that n-1 a2-1 an- det (V) II (aj - αi) 1<i<j<n n L(z) = Σ Bk (II k=1 for general n. (Hint: Use elementary column operations to reduce the first row to (1,0,0...), then expand by the first row.) (iii) Consider the following interpolation linear map Þ: K[2]<(n-1) →→→→K", Þ(f(z)) = (f(ai)). Compute its matrix in the basis 1, 2,..., 2-1 of the polynomial space and the standard basis of Kn. (iv) Verify that the Lagrange interpolation polynomial 2- αi Ok 1≤i≤n, itk α₂ is a solution of the interpolation problem Þ(L(z)) = (B₂). (v) Use Cramer's rule (from Linear Algebra) and parts (ii), (iii), (iv) of this problem to derive a formula for the coefficients of L(z). (Warning: do not try to compute the determinant in the numerator after you applied Cramer's rule or to simplify the answer.) Q3. Let a1, 02,..., an be distinct elements of a field K. Consider the Vandermonde matrix 1 αι αξ 1 -(45) 1 an an V := (i) Compute the determinant det (V) when n = 2. (ii) Use induction to show that n-1 a2-1 an- det (V) II (aj - αi) 1<i<j<n n L(z) = Σ Bk (II k=1 for general n. (Hint: Use elementary column operations to reduce the first row to (1,0,0...), then expand by the first row.) (iii) Consider the following interpolation linear map Þ: K[2]<(n-1) →→→→K", Þ(f(z)) = (f(ai)). Compute its matrix in the basis 1, 2,..., 2-1 of the polynomial space and the standard basis of Kn. (iv) Verify that the Lagrange interpolation polynomial 2- αi Ok 1≤i≤n, itk α₂ is a solution of the interpolation problem Þ(L(z)) = (B₂). (v) Use Cramer's rule (from Linear Algebra) and parts (ii), (iii), (iv) of this problem to derive a formula for the coefficients of L(z). (Warning: do not try to compute the determinant in the numerator after you applied Cramer's rule or to simplify the answer.)

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