MATHEMATICAL TRIPOS Part IA Thursday 27 May 2010 9:00 am to 12:00 noon PAPER 1 Before you begin read these instructions carefully. The examination paper is divided into two sections. Each question in Section II carries twice the number of marks of each question in Section I. Candidates may attempt all four questions from Section I and at most five questions from Section II. In Section II, no more than three questions on each course may be attempted. Complete answers are preferred to fragments. Write on one side of the paper only and begin each answer on a separate sheet. Write legibly; otherwise you place yourself at a grave disadvantage. At the end of the examination: Tie up your answers in separate bundles, marked A, B, C, D, E and F according to the code letter affixed to each question. Include in the same bundle all questions from Section I and II with the same code letter. Attach a completed gold cover sheet to each bundle. You must also complete a green master cover sheet listing all the questions you have attempted. Every cover sheet must bear your examination number and desk number. STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS Gold Cover sheets None Green master cover sheet You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.2 SECTION I 1A Vectors and Matrices Let A be the matrix representing a linear map : Rn ! Rm with respect to the bases {b1, . . . , bn} of Rn and {c1, . . . , cm} of Rm, so that (bi) = Ajicj . Let {b′1, . . . , b′n} be another basis of Rn and let {c′1, . . . , c′m} be another basis of Rm. Show that the matrix A′ representing with respect to these new bases satisfies A′ = C−1AB with matrices B and C which should be defined. 2C Vectors and Matrices (a) The complex numbers z1 and z2 satisfy the equations z 3 1 = 1 , z 9 2 = 512 . What are the possible values of |z1 − z2|? Justify your answer. (b) Show that |z1 + z 2| 6 |z1| + |z 2| for all complex numbers z1 and z 2 . Does the inequality |z1 +z 2|+|z1−z 2| 6 2max (|z1|, |z 2|) hold for all complex numbers z1 and z 2? Justify your answer with a proof or a counterexample. 3D Analysis I Let Pn>0 anzn be a complex power series. State carefully what it means for the power series to have radius of convergence R, with R 2 [0,1] . Suppose the power series has radius of convergence R, with 0 < R < 1. Show that the sequence | anzn| is unbounded if |z| > R. Find the radius of convergence of Pn>1 zn/n3. Part IA, Paper 13 4E Analysis I Find the limit of each of the following sequences; justify your answers. (i) 1 + 2 + . . . + n n2 ; (ii) npn ; (iii) (an + bn)1/n with 0 < a 6 b . Part IA, Paper 1 [TURN OVER4 SECTION II 5A Vectors and Matrices Let A and B be real n × n matrices. (i) Define the trace of A, tr (A), and show that tr ATB= tr BTA. (ii) Show that tr ATA> 0 , with tr ATA= 0 if and only if A is the zero matrix. Hence show that tr(ATB)2 6 tr ATAtr BTB. Under what condition on A and B is equality achieved? (iii) Find a basis for the subspace of 2 × 2 matrices X such that tr ATX= tr BTX= tr CTX= 0 , where A = 1 1 2 0 , B = 1 1 0 −2 , C = 0 0 1 1 . 6C Vectors and Matrices Let a1, a2 and a3 be vectors in R3. Give a definition of the dot product, a1 · a2 , the cross product, a1 ×a2 , and the triple product, a1 · a2 ×a3 . Explain what it means to say that the three vectors are linearly independent. Let b1, b2 and b3 be vectors in R3. Let S be a 3×3 matrix with entries Sij = ai · bj . Show that (a1 · a2 × a3)(b1 · b2 × b3) = det(S) . Hence show that S is of maximal rank if and only if the sets of vectors {a1, a2, a3} and {b1, b2, b3} are both linearly independent. Now let {c1, c2, . . . , cn} and {d1, d2, . . . , dn} be sets of vectors in Rn, and let T be an n × n matrix with entries Tij = ci · dj . Is it the case that T is of maximal rank if and only if the sets of vectors {c1, c2, . . . , cn} and {d1, d2, . . . , dn} are both linearly independent? Justify your answer with a proof or a counterexample. Given an integer n > 2, is it always possible to find a set of vectors {c1, c2, . . ., cn} in Rn with the property that every pair is linearly independent and that every triple is linearly dependent? Justify your answer. Part IA, Paper 15 7B Vectors and Matrices Let A be a complex n×n matrix with an eigenvalue . Show directly from the definitions that: (i) Ar has an eigenvalue r for any integer r > 1; and (ii) if A is invertible then 6= 0 and A−1 has an eigenvalue −1. For any complex n×n matrix A, let A(t) = det(A−tI). Using standard properties of determinants, show that: (iii) A2(t2) = A(t) A(−t) ; and (iv) if A is invertible, A−1(t) = (detA)−1(−1)ntnA(t−1) . Explain, including justifications, the relationship between the eigenvalues of A and the polynomial A(t). If A4 has an eigenvalue µ, does it follow that A has an eigenvalue with 4 = µ? Give a proof or counterexample. 8B Vectors and Matrices Let R be a real orthogonal 3×3 matrix with a real eigenvalue corresponding to some real eigenvector. Show algebraically that = ±1 and interpret this result geometrically. Each of the matrices M = 0@ 0 1 0 0 0 1 1 0 01A, N = 0@ 1 −2 −2 0 1 −2 0 0 11A, P = 13 0@ 1 −2 −2 −2 1 −2 −2 −2 11A has an eigenvalue = 1. Confirm this by finding as many independent eigenvectors as possible with this eigenvalue, for each matrix in turn. Show that one of the matrices above represents a rotation, and find the axis and angle of rotation. Which of the other matrices represents a reflection, and why? State, with brief explanations, whether the matrices M, N, P are diagonalisable (i) over the real numbers; (ii) over the complex numbers. Part IA, Paper 1 [TURN OVER6 9E Analysis I Determine whether the following series converge or diverge. Any tests that you use should be carefully stated. (a) Xn>1 n! nn ; (b) Xn>1 1 n + (log n)2 ; (c) Xn>1 (−1)n 1 + pn ; (d) Xn>1 (−1)n n (2 + (−1)n) . 10F Analysis I (a) State and prove Taylor’s theorem with the remainder in Lagrange’s form. (b) Suppose that e : R ! R is a differentiable function such that e(0) = 1 and e′(x) = e(x) for all x 2 R. Use the result of (a) to prove that e(x) =Xn>0 xn n! for all x 2 R. [No property of the exponential function may be assumed.] Part IA, Paper 17 11D Analysis I Define what it means for a bounded function f : [a,1) ! R to be Riemann integrable. Show that a monotonic function f : [a, b] ! R is Riemann integrable, where −1 < a < b < 1. Prove that if f : [1,1) ! R is a decreasing function with f(x) ! 0 as x ! 1, then Pn>1 f(n) and R 1 1 f(x) dx either both diverge or both converge. Hence determine, for 2 R, when Pn>1 nconverges. 12F Analysis I (a) Let n > 1 and f be a function R → R. Define carefully what it means for f to be n times differentiable at a point x0 ∈ R. Set sign(x) = (x/|x|, x 6= 0, 0, x = 0. Consider the function f(x) on the real line, with f(0) = 0 and f(x) = x2 sign(x) cos x, x 6= 0. (b) Is f(x) differentiable at x = 0? (c) Show that f(x) has points of non-differentiability in any neighbourhood of x = 0. (d) Prove that, in any finite interval I, the derivative f′(x), at the points x ∈ I where it exists, is bounded: |f′(x)| 6 C where C depends on I. END OF PAPER Part IA, Paper 1