A. MillerM340 Final ExamMay 14, 1990There is a total of 35 points. Put your answers on separatesheets of paper. Keep this exam, only hand in the answers. Youcan pick up your graded exam on Wednesday, VV403, 2-3. Yourgrade will be posted on the door.1. (3 points) Find a diagonal matrix D and invertible matrix P such thatA = P DP−1whereA ="1 −21 4#2. (6 points) Find bases for the kernel(A) and the range(A) whereA =1 2 −2 −3 2−1 −2 3 4 03 6 −3 −6 −33. (3 points) Find an orthonormal sequence u1, u2, u3so that span{v1, . . . , vk} =span{u1, . . . , uk} for k = 1, 2, 3.v1=010v2=−324v3=4−834. (3 points) Find the inverse of the matrix A:A =1 2 01 3 1−1 −1 25. (3 points) Find the determinant of the matrix A:A =1 0 0 0 0−17 0 3 0 −856 −1 0 0 0−4 0 0 1 012 3 0 0 −216. (4 points) Suppose that A ∈ Fn×m, B ∈ Fm×p, C ∈ Fp×r.Prove that (AB)C = A(BC).7. (4 points) If W1and W2are subspaces of a vector space V , then defineW1MW2= {w1+ w2: w1∈ W1, w2∈ W2}.Prove that W1LW2is a subspace of V .8. (4 points) Suppose that u1, u2is an orthonormal sequence in an innerproduct space and c1, c2∈ R. Prove that ||c1u1+ c2u2|| =qc21+ c22.9. (4 points) Suppose that U ∈ Rn×nis an upper triangular matrix, UTisits transpose, and UTU = UUT. Prove that U is a diagonal matrix.10. (1 points) True or False: Suppose that u1, u2, . . . , unis an orthonormalsequence, then u1, u2, . . . , unare linearly independent.2Answers1. p = d =1.000 -2.000 2.000 0.000-0.500 2.000 0.000 3.0002. basis of range: 1.000 -2.000 2.000-1.000 3.000 0.0003.000 -3.000 -3.000basis of kernel:-2.000 1.0001.000 0.0000.000 -1.0000.000 1.0000.000 0.0003. u1 u2 u30.000 -0.600 0.8001.000 0.000 0.0000.000 0.800 0.6004. 7.000 -4.000 2.000-3.000 2.000 -1.0002.000 -1.000 1.0005. -636.entryij((AB)C) = Σkentryik(AB)entrykj(C) =ΣkΣlentryil(A)entrylk(B)entrykj(C)= Σlentryil(A)Σkentrylk(B)entrykj(C)= entryij(A(BC))7. c1(w1+ w2) + c2(w01+ w02) = (c1w1+ c2w01) + (c1w2+ c2w02)8. ||c1u1+c2u2||2= hc1u1+c2u2, c1u1+c2u2i use linearity of inner productand hu1, u2i = 0 and hu1, u1i = hu2, u2i = 1.9. Use that entryii(U ∗ UT) = ||rowi(U)||2, entryii(UT∗ U) = ||coli(U)||2and look at minimal row i with a nonzero offdiagonal entry.10.