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 Va = LU where 0 a2 - 01 0 L = as - a1 (as - a1) (as - a2) a4 - a1 (as - a1) (as - 12) (as - a1)(a4 - 42)(as - a3) and 1 as U = 1 a taz a; + aja2 + az 0 a1 + a2 + as 0 O (b) Use Part (a) to compute the determinant of VA. (c) Show that if a1, a2, as and a4 are distinct real numbers then there exists a unique polynomial of degree at most three, p(r), satisfying p(a1) = bi, p(az) = b2, p(as) = bs and p(as) = ba, for any b1, b2, bs, ba E R. Read Polynomial Interpolation (Page 163-166) of textbook. 61 6 (d) Let b3 = 6 ba O Find the unique polynomial p(r) of degree at most three satisfying p(2) = b1, p(1) = b2, p(0) = bs and p(-1) = b4