Please solve a, b, c

1. Given n real numbers a1, . .., an, the Vandermonde matrix is the n x n matrix Vn whose (r, s)-entry is a$ 1 for r, s = 1, ...,n. [10 marks (a) Show that V4 = LU where 0 O O 1 a2 - a1 L = 1 a3 - a1 (a3 - a1) (a3 - a2) a4 - a1 (a4 - a1) (a4 - a2) (a4 - a1) (a4 - a2) (a4 - a3) and a1 as 0 1 a1 + a2 U = ai + ala2 + a? 0 a1 + a2 + a3 O O (b) Use Part (a) to compute the determinant of V4. (c) Show that if a1, a2, a3 and a4 are distinct real numbers then there exists a unique polynomial of degree at most three, p(x), satisfying p(al) = b1, p(a2) = b2, p(as) = b3 and p(a4) = b4, for any b1, b2, b3, b4 E R. Read Polynomial Interpolation (Page 163-166) of textbook