The n × n Vandermonde matrix associated with a set of points {xo,...,xn−1} = R is Vn(xo,...,xn-1) = 1 x1 xn-1 x² The Vandermonde matrix appears frequently when approximating a function whose value is only known at the points {xo,...,xn-1}. (We have already encountered two variants of Vn(x0,...,xn−1) in earlier assignments!) (a) Using elementary row and column operations together with the properties of the deter- minant, show that det Vn (x0,...,xn−1) = det Vn (x0,...,xn−1), where 0 0 0 0 x1x0 0 x2-x0 x1(x1 - x0) x2(x2-x0) (x2-x0) Vn(xo,...,xn−1) = 0 : x-(xn−1x0)] xn-1x0 Xn-1(xn−1 - x0) (Hint: First introduce the zeros in the first column using row operations. Then, use column operations to introduce the zeros in the top row: be strategic about how you introduce these zeros.) (b) Use your result from part (a) to establish the formula - - - det Vn(xo,...,xn−1) = (x1 − x0)(x2 − x0)... (xn−1 − x0) det Vn-1(x1,...,xn−1). (c) Apply the result in part (b) recursively to deduce that det Vn (xo,,xn-1) = Пo<i<j≤n-1(xj — xi). - - Here, the product notation Пo<i<j≤n-1(xj − xi) means to multiply all the differences (x, x) such that i and j are between 0 and n - 1 and i < j. -