Name Class TimeMath 2270 Extra Credit ProblemsChapter 5November 2008Due date: See the internet due date for 7.6, which is the due da te for these problems. Records are locked on that dateand only corrected, never appended.Submitted work. Please submit one stapled package per problem. Kindly label problemsExtra Credit . Labeleach problem with its corresponding problem number. You may attach this printed sheet to simplify your work.Problem Xc5.1-10. (Angle)For which values of k are the vectors u =2k13and v =1k2perpendicular?Problem Xc5.1-26. (Orthogonal Projection)Find the orthogonal projection of w onto the s ubspace V , givenw =1000, V = span1111,11−1−1,1−1−11.Problem Xc5.1-34. (Minimization)Among all the vectorsxyzin R3, find the one with unit length that minimizes the sum x + 2y + 3z.Problem Xc5.2-14. (Gram-Schmidt Basis)Given the basis below, lab e le d v1, v2, v3, find the Gram-Schmidt basis u1, u2, u3.1717,0727,215213.Problem Xc5.2-20. (QR-Factorization)Find the factorization M = QR, givenM =4 25 00 0 −23 −25 0.Problem Xc5.2-34. (Kernel)Find an orthonormal basis for the kernel of the matrix A =1 1 1 10 1 2 3.Problem Xc5.2-38. (QR-Factorization)Find the factorization M = QR, given M =0 −3 00 0 04 0 00 0 5Problem Xc5.3-11. (Orthogonal Matrices)Given A and B are orthogonal, then which of the following must be orthogonal?(a) 2A, (b) ABA, (c) A−1BT, (d) A − AB, (e) AB + BA, (f) −BAProblem Xc5.3-20. (Symmetric Matrices)Given A and B are symmetric matrices and A is invertible, then which o f the following must also be symmetric?(a) ATA, (b) ABA, (c) A−1B, (d) A − B, (e) A − BA, (f) A − AT, (g) ATBTBA, (h) B(A + AT)BTProblem Xc5.3-26. (Dot Product)Let T be an orthogonal transformation from Rnto Rn. Prove that u · v = T (u) · T (v) for all vectors u and v in Rn.Problem Xc5.3-32a. (Orthogonal Matrices)Assume A is n × m and ATA = I. Is AATthe identity matrix ? Explain.Problem Xc5.3-44. (Orthogonal Matrices)Consider an n × m matrix A. Find in terms of n and m the value o f the sum rank(dim(A)) + rank(ker(AT)).Problem Xc5.3-50. (QR-Factorization)(a) Find all square ma trices A that are b oth orthogonal and upper triangular with positive diagonal entries.(b) Show that the QR-factorization is unique for an invertible square matrix A. Hint: see Ex ercise 50b in Bretscher3E, section 5.3.Problem Xc5.4-5. (Basis of V⊥)Find a basis for V⊥, where V = ker(A) andA =1 1 1 11 2 4 30 0 0 0.Problem Xc5.4-16. (Rank)Prove or disprove: The equation rank(A) = rank(ATA) hold for all square matrices A.Problem Xc5.4-22. (Least Squares)Find the least sq uares solution x∗of the system Ax = b, givenA =3 25 34 5, b =10184.Problem Xc5.4-26. (Least Squares)Find the least sq uares solution x∗of the system Ax = b, givenA =1 2 34 5 67 8 9, b =200.Problem Xc5.5-10. (Orthonormal Basis)Find an orthonormal basis for V⊥, where V = span{1 + t2}, in the space W of a ll polynomials a0+ a1t + a2t2withinner pro duct < f, g >=12R1−1f(t)g(t)dt.2Problem Xc5.5-24. (Orthonormal Basis)Consider the linear space P of all polynomials with inner product < f, g >=R10f(t)g(t)dt. Let f, g, h be threepolynomials satisfying the relations< f, f >= 4 < f, g >= 0 < f, h >= 8< g, f >= 0 < g, g >= 2 < g, h >= 4< h, f >= 8 < h, g >= 4 < h, h >= 10(a) Find < f, g + 2h >.(b) Find kg + hk.(c) Find c1, c2satisfying projspan{f,g}(h) = c1f + c2g.(d) Find an orthonormal basis for the span of f, g, h expressed as linear combinations of f , g and h.End of extra credit problems chapter