MATH 5610/6860FINAL EXAM PRACTICE PROBLEMS #21. Find the LU factorization (with L being a unit lower triangular matrix) of thematrixA =1 1 21 2 11 1 12. Find the LU factorization (with L being a unit lower triangular matrix) of thematrixA =1 1 22 1 −11 0 −1.(a) Without pivoting.(b) With scaled row pivoting. Clearly indicate which rows have been permuted.3. (K&C 4.5.42) Consider the matrixA =1 21 2.01.(a) Compute the condition number κ∞(A).(b) Verify that the bound for the relative error given by the condition numberis satisfied with the right hand sides b = (4, 4)Tandeb = (3, 5)T.4. (K&C 4.6.8) Show that if the i−th equation of the linear system Ax = b isdivided by aiiand then Richardson’s method is applied to solve the system, theresult is the same as applying Jacobi’s method.5. Find the Legendre polynomial p5(x) from the two previous Legendre polynomialsp3(x) and p4(x).p3(x) = x3−35xp4(x) = x4−67x2+335p5(x) = x5−109x3+521.6. Find the polynomial of the form p(x) = ax+b that best approximates f(x) = x3in [0, 1], where the norm is induced by the product(f, g) =Z10f(x)g(x)dx.12 MATH 5610/6860 FINAL EXAM PRACTICE PROBLEMS #27. Let xj= 2πj/N and Ej(x) = exp[2iπjx/N ], j = 0, . . . , N − 1. Recall thepseudo-inner product(f, g)N=1NN−1Xj=0f(xj)g(xj),and its fundamental property(En, Em) =(1 if n − m is divisible by N0 otherwise.Consider the trigonometric polynomialp =N−1Xk=0ckEk.Show the discrete Parseval’s identityN−1Xj=0|p(xj)|2= NN−1Xj=0|ck|2.Hint: For j = 0, . . . , N − 1 we have,|p(xj)|2= N−1Xk=0ckEk(xj)!