Notational Convention. Let fej e R" | je N[1, n]} be the standard unit basis of R", Le., e] = (1, 0, 0, . . . , 0), e2 = (0, 1, 0, ... , 0)', ..., en = (0, 0, 0, ... , 1). In this problem, we denote by G; the Gaussian transformation matrix of the form Gj=It NajeeJ. i=j+1 In addition, let P(i, j) e Roxx be the elementary permutation matrix obtained by interchanging the ith and the jth rows of the same-sized identity matrix. Questions. (a) (7 points) Let 1 j bijee], where i=j+1 if i # k and i # 1, bij = if i = k. akj, ifi = e. (b) (7 points) Show that Gal = 1 - _ aijereJ. i=j+1 (c) (7 points) Let j cakerez. i=j+1 i=k+1 (d) (9 points) Use the previous parts to find PLU factorization, PA = LU, by hand. A =