Homework Assignments for Math 545 Section 1

• Assignment 1   Due Thursday, September 16.
  • 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.
• Assignment 2   Due Thursday, September 22.
  • Read Sections 11, 12
  • Section 11 page 87: 6 (a), (d), 7 (use the definition of rank given in Definition 8.6 page 56), 8 (b), (c), 9, 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.
• Assignment 3   Due Thursday, September 29..
  • Section 13 page 107: 2, 3, 4, 5, 7, 8, 10, 11, 12
  • Start working on the extra problem on Gaussian curvature for section 13, which is due with Assignment 4.
• Assignment 4   Due Thursday, October 6.
  • Section 15 page 129: 1 (a), (b), 2, 3 (a), (b), 4, 5
  • Solve Question 4 in Fall 2010 Midterm 1.
  • Extra problem on Gaussian curvature for section 13.
• Assignment 5   Due Thursday, October 13.
  • 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.
• Assignment 6   Due Thursday, October 27.
  • 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, 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).
• Assignment 7   Due Thursday, November 3.
  • 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.
  • 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 Thursday, November 10.
  • Section 23 page 201: 1, 2, 3, 4, 5, 6.
• Assignment 9   Due Thursday, December 1.
  • Extra problem on the Primary Decomposition Theorem.
  • 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.
  • Note that part g was added to problems 1, 2, and the extra problem above.
• Assignment 10   Due Thursday, December 8.
  • Section 24 page 215: 8, 9, 10 a
  • Section 34 page 304: (material will be covered on Tuesday, 11/30) 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 11   Due Thursday, December 15 (not to be handed in, but material was covered in the last week and will be covered in the final.
  • 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.
  • Section 25 page 225: 1 a, b, 2, 3, 4, 5 (see hint below), 6 a, b, 7, 8
  • Hint for problem 5: Break the proof into three steps:
    Step 1: Prove that V is cyclic if and only if each of the summands V_i, in the Primary Decomposition of V with respect to T, is cyclic. Here you will need to use the fact that V_i=E_i(V), and that the eidempotent E_i is a polynomial in T. For example, if V=span{v,T(v), T^2(v), ...} and we set v_i:=E_i(v), then it is easy to show, that V_i=span{v_i,T(v_i), T^2(v_i), ...}. Conversely, if each V_i is cyclic, generated by v_i, show that V is cyclic generated by v_1+ ... +v_r.
    Step 2: Assume that the minimal polynomial m(x) of T is a prime power, m(x)=p(x)^e, where p(x) is a prime polynomial. Show that the degree of m(x) is equal to the dimension of V, if and only if V 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).
    Step 3: Combine the above two steps, to prove the statement.
  • Section 25: Additional problems.