Homework Assignments for Math 545 Section 1

• Assignment 1   Due Friday, February 3.
  • Read Sections 2, 3, 4, 5, 6, 7, 8, 9, 11 (Review of material from math 235)
  • Section 2 page 15: 2 (a), 4
  • Section 3 page 25: 6, 9, 10
  • Section 4 page 33: 3 (b), (c), (g) with *fixed* coefficients alpha, beta, and gamma, (h), (Justify your answer by verifying the conditions in definition 4.1), 4 (f), (g), 7, 9
  • Section 5 page 37: 3, 5
  • Section 6 page 48: 3 (e), 4, 5 (a)
  • Section 7 page 52: 1, 5, and the following modification of problem 4. Let S and T be both three dimensional subspaces of R_4. What are all the possible dimentions of their intersection?
  • Section 9 page 68: 1, 4 (a), (c)
  • Section 11 page 87: 1, 3, 4, 6 (a), (d), (questions 7, 8, 9, and 10, were postponed to next week).
• Assignment 2   Due Friday, February 10.
  • Section 11 page 87: 7 (use the definition of rank given in Definition 8.6 page 56), 8 (b), (c), 9 (in part c there is a typo, F_n should be R_n. Hint: use the material in section 11 as well as Corollary 9.4 page 64), 10.
  • Section 12 page 98: 1 (first two), 4, 5, 7 (a), (b), (c), 8 (Assume that the equation Ax=b has a unique solution, for every vector b).
  • Solve Question 2 in Fall 2010 Midterm 1.
  • Section 13 page 107: 2, 3, 4, 5, 7, 8, 10, 11, 12
  • Extra problem for section 13: Solve question 5 in Spring 2012 midterm 1.
  • Start working on the extra problem on Gaussian curvature for section 13, which will be due on February 24. It will receive a separate grade of weight equal to a homework assignment.
• Assignment 3   Due Wednesday, February 22.
  • Section 15 page 129: 1 (a), (b), 2, 3 (a), (b), 4, 5
  • Solve Question 4 in Fall 2010 Midterm 1.
  • Continue working on the extra problem on Gaussian curvature for section 13, which will be due on Friday February 24. It will receive a separate grade of weight equal to a homework assignment.
• Assignment 4   Due Friday, February 24.
  • extra problem on Gaussian curvature for section 13.
• Assignment 5   Due Monday, February 27.
  • Section 15 page 129: 7 (see Definition 11.10, and note that by Theorem 11.11 it suffices to show that a) the product of two orthogonal transformations is an orthogonal transformation, b) the inverse of an orthogonal transformation is an orthogonal transformation.),
    8, 9, 11, 13.
  • Section 15 page 129: 10 Hint: Choose bases {v_1, ..., v_k} of W_1 and {w_1, ..., w_k} of W_2. Use Theorem 7.4 to extend each to a basis of V. Next use Theorem 15.9 and Theorem 15.11 parts (1) and (4). Carefully explain what property of the Gram-Schmidt process ensures that T(W_1)=W_2.
  • Solve Question 3 in Fall 2011 Midterm 1.
• Assignment 6   Due Friday, March 9.
  • Section 16 page 139: 1 (b), (c) (use row reduction as in Example A), 3
  • Section 18 page 149: 1, 2, 5, 6.
  • Additional problem for section 18:
    • (a) Let R be a rotation of the plane with angle a about the origin. Compute det(R).
    • (b) Let V be an n-dimensional vector space with an inner product and u a unit vector in V. The reflection S of V with respect to u (or more precisely, with respect to the subspace orthogonal to u) is given by
      S(v) = v - 2(v,u)u.
      (i) Prove that S is a linear transformation.
      (ii) Prove that S is an orthogonal transformation.
      (iii) Prove that det(S)=-1.
  • Section 20 page 175: 1, 2, 4, 5, 6, 7a, c, d (for parts c and d factor first over the complex numbers, then over the real numbers, and then over the rational numbers. Note also that the roots of x^n-a^n=0 are all of the form: a[n-th root of unity]. Hence, once you know one n-th root a of x^n-b, you know them all).
  • Section 21 page 182: 1, 2, 3, 4, 6 (see Definition 11.10).
  • Extra problem for section 21: Solve problem 3 in Spring 2012 Midterm 2.
  • Section 22: Let V be an n-dimensional vector space with an inner product and u a unit vector in V. Recall, that the reflection R of V, with respect to the subspace orthogonal to u, is given by
    R(v) = v - 2(v,u)u.
    Find the minimal polynomial of R.
• Assignment 7   Due Friday, March 16.
  • Section 22 page 192: 6b modified: Let V be a vector space over a field F, T:V->V a linear transformation, v an eigenvector of T with eigenvalue t, and
    g(x)=c_n x^n + ... + c_1 x + c_0
    a polynomial in F[x]. Show the following equality of vectors in V
    (g(T))v=g(t)v.
    The left hand side is obtained by evaluating first g(x) on T, to get a linear transformation g(T), and then applying g(T) to the vector v. The right hand side is obtained by evaluating first g(x) on the scalar t, to get a scalar g(t), and then scalar multiplying v.
  • Section 22 page 192: 3, 4 (Hint: use problem 6b above),
    5, 6a, 8, 9, 10, 12.
• Assignment 8   Due Friday, March 30.
  • Section 23 page 201: 1, 2, 3, 4, 5, 6.
  • Extra problem on the Primary Decomposition Theorem.
• Assignment 9   Due Friday, April 6.
  • Solve all the problem in Fall 2010 Midterm 2.
  • Solve Problem 6 from Spring 2007 Midterm 2.
• Assignment 10   Due Friday, April 13.
  • Section 24 page 215: 1, 2, 3, 4, 7
  • Section 24: Problem 6 modified. Let T be a linear transformation of a vector space V, over the field of complex numbers, such that T^n=cT (cT is equal to the n-th power of T), for some complex number c. Discuss whether or not there exists a basis of the vector space V consisting of eigenvectors of T.
  • Section 24 page 215: Additional problem: Let T be a linear transformation on a vector space over the complex number field with a basis u, v, w, such that
    T(u) = u-v
    T(v) = u+3v
    T(w) = -u-4v-w.
    Answer the questions in Excercise 1 page 215 for this T.
  • Section 24 page 215: 8, 9, 10 a
• Assignment 11   Due Friday, April 20.
  • Section 34 page 304: 1, 3, 4, 7 (see hint below), 8, 9 (only the second matrix).
  • Hint for problem 7 of section 34: Method 1: (the shortest) Let M be the 2x2 matrix of the associated system. Calculate e^{tM} directly from its definition, by summing up separately the even powers of tM and the odd powers of tM, using the identity M^2=-(m^2)I.
    Method 2: You can also diagonalize the matrix over the complex numbers, obtain the general complex valued solution, and show that the solution to the initial valued problem is real valued. This method is longer than the ad hoc method 1, but for more general systems of linear O.D.E's, with diagonalizable matrices with complex eigenvalues, this is the only method available.
  • Section 25 page 225: To be announced.
• Assignment 12   Due Friday, April 27.
  • Section 25: Additional problems:
    • Let V be the real plane R^2 and A the matrix
      cos(t)   -sin(t)
      sin(t)    cos(t)
      of the rotation of the plane an angle t radians counterclockwise. Assume 0 <= t < 2pi. Let v=e_1 be the first column of the 2x2 identity matrix. Calculate the order m_v(x) of v with respect to A (see Definition 25.5). Calculate also the minimal polynomial of A. Note, you will need to treat the cases t=0 and t=pi seperately.
    • Let T be a linear transformation on a vector space over the real numbers with a basis u, v, w, such that
      T(u) = u-v
      T(v) = u+3v
      T(w) = -u-4v-w.
      i) Find the order m_v(x) of v. Hint: Read the proof of Lemma 25.4 for an algorithm for finding the order of v.
      ii) Let z=v+w. Find the order m_z(x) of z.
  • Read Definition 25.15 and 25.17 (in class we covered the material up to definition 25.14, but you will need these later definitions for the homework). Note that part of the homework was postponed as Assignment 13.
  • Section 25 page 225: 1 a, b, 2, 3, 4, 5 (see hint below).
  • Hint for problem 5: One directions is easier: If V is cyclic, then the minimal polynomial of T is equal to its characteristic polynomial. If you can not do it on your own, check the answer at the end of the book. The other direction is not proven at the end of the book. For the other direction argue as follows. Let the minimal polynomial of T be m(x)=p_1(x)^{e_1} ... p_s(x)^{e_s}. Let V_i:=ker(p_i(T)^{e_i}) be the summand in the primary decomposition of V with respect to T. The general strategy is:
    (a) The minimal polynomial is equal to the characteristic polynomial of T =>
    (b) The degree of the minimal polynomial is equal to the dimension of V =>
    (c) The degree of the minimal polynomial of the restriction of T to each summand V_i is equal to the dimension of V_i =>
    (d) Each summand V_i is cyclic generated by some vector w_i with respect to T =>
    (e) V is cyclic, generated by the vector v=w_1+ ... w_s.
    Some hints for the proof of the above implications:
    (b)=>(c) Use the Cayley-Hamilton Theorem for the restriction of T to V_i.
    (c)=>(d): The minimal polynomial m_i(x), of the restriction T_i of T to the summand V_i in the primary decomposition, is a prime power, m_i(x)=p(x)^e, where p(x) is a prime polynomial. Show that if the degree of m_i(x) is equal to the dimension of V_i, then V_i is cyclic. Here you will use the fact, that the dimension of a cyclic subspace generated by v is equal to the degree of the order m_v(x).
    (d)=>(e): Set f_i(x)=m(x)/p_i(x). Show that f_i(T)(v) is not zero. Conclude that the order m_v(x) does not divide f_i(x). Use the fact that the order m_v(x) divides m(x) to conclude the equality m_v(x)=m(x).
• Assignment 13   Due Monday, April 30.
  • Section 25 page 225: 6 a, b, 7, 8.
  • Section 25: Additional problems.
  • Section 25: Use the following algorithm in order to solve the following problem.