Homework Assignments for Math 545 Section 1

• Assignment 1   Due Friday, February 2.
  • Read Sections 2, 3, 4, 5, 6 (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 (h), 7
  • Section 5 page 37: 3, 5
  • Section 6 page 48: 3 (e), 4, 5 (a)
• Assignment 2   Due Friday, February 9.
  • Read Sections 7, 8, 9, 11, 12 (Review of material from math 235)
  • Section 7 page 52: 1, 4, 5
  • Section 9 page 68: 1, 4 (a),
  • Section 11 page 87: 1, 3, 4, 6 (a), (d), 7, 8 (b), (c), 10
  • Section 12 page 98: 1 (first two), 4, 5, 7 (a), (b), (c)
• Assignment 3   Due Friday, February 16.
  • Section 11 page 87: 9
  • Section 12 page 98: 8 (Assume that the equation Ax=b has a unique solution, for every vector b).
  • Section 13 page 107: 2, 3, 4, 5, 7, 8, 10, 11
• Assignment 4   Due Friday, March 2.
  • Section 15 page 129: 1 (a), (b), 2, 3 (a), (b), 4, 5, 7, 8, 9, 11, 13.
• Assignment 5   Due Friday, March 9.
  • Section 13 page 107: 12
  • 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 21 page 182: 1, 2, 3, 6.
• Assignment 6   Due Monday, March 12. (It will not be collected, nor graded, but the material of section 21 is included in the midterm).
  • Section 21 page 182: 4.
  • Let F be a field. A polynomial f in F[z] is called a prime (or irreducible) polynomial, if the degree d of f is positive, and f can NOT be factored as a product f=gh, where g,h are both polynomials in F[z] of positive degree. The Fundamental Theorem of Algebra means, that when F is the field C of complex numbers, a polynomial f in C[z] is prime, if and only if the degree of f is 1.
    • (a) Let F=R be the field of real numbers. Factor z^5-1 as a product of prime polynomials in R[z].
    • (b) Let F=Q be the field of rational numbers. Factor z^4-4 as a product of prime polynomials in Q[z]. (Recall, that the square root of 2 is not a rational number).
• Assignment 7   Due Wednessday, March 28.
  • Hand in assignment 6 problems as well.
  • 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, April 6.
  • 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 23 page 201: 1, 2, 3, 4, 5, 6
• Assignment 9   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.
• Assignment 10   Due Friday, April 20.
  • 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.
    i) Answer the questions in Excercise 1 page 215 for this T.
    ii) Let A be the matrix of T in the basis u, v, w. Calculate the k-th power A^k, for all positive integers k, using the Jordan decomposition of T you found.
  • Section 24 page 215: 8, 9, 10 a
• Assignment 11   Due Monday, April 23.
  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.
• Assignment 12   Due Friday, May 4.
  • 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.
• Assignment 13   Due Friday, May 11.
  • 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 30 page 270: 1, 2, 3, and the Additional problems.
  • Section 31 page 275: To be announced
• Assignment 14   Due Friday, May 18 (not to be handed in, but material will be covered in the final exam).
  • Section 31 page 275: 1, 2 a, b (include a sketch of the curve Q(x_1,x_2)=10 for both a and b), 3, 4, 9