Spring 2018

Math 3110 Linear Algebra for Engineers

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Homework 7

Due Friday May 4, in class
  • Solution Manual: Chapter 7; Chapter 8.
  • Section 7.3
    • In class: 3*:
      • 3*. Follow the directions for Problem 3 in Problem Set 7.3, with \[A_0 = \left[\matrix{-1&0&4\cr 0&1&2}\right].\]
    • Homework: 1, 3-5
    • Hand In: 3**, 4*:
      • 3**. Follow the directions for Problem 3 in Problem Set 7.3, with \[A_0 = \left[\matrix{-1&3&4\cr -3&-1&1}\right].\]
      • 4*. Follow the directions for Problem 4 in Problem Set 7.3, with \(A = \left[\matrix{2&2&1&0\cr 1&-1&0&1\cr 1&0&-2&-1}\right]\).
  • Section 7.4
    • In class: 5*-9*:
      • 5*-9*. Follow the directions for Problems 5-9 in Problem Set 7.4, with \[A = \left[\matrix{2&2\cr -1&1}\right].\]
    • Homework: 1-4, 5-9, 10-11, 15
    • Hand In: 5**-9**, 10*-11*:
      • 5-9**. Follow the directions for Problems 5-9 in Problem Set 7.4, with \[A = \left[\matrix{4&4\cr -1&1}\right].\]
      • 10-11*. Follow the directions for Problems 10-11 in Problem Set 7.4, with \(A = \left[\matrix{-2&2&1}\right]\).
  • Section 8.1
    • Watch Lecture 30: Linear Transformations and their matrices of the MIT OpenCourseWare series.
    • In class: 8*:
      • 8*. Find the range and kernel (like the column space and nullspace) of \(T\):
        • (a) \(T(v_1,v_2) = (0,v_1+v_2)\)
        • (b) \(T(v_1,v_2,v_3) = (v_2,v_3)\)
        • (c) \(T(v_1,v_2) = (v_2,0)\)
        • (d) \(T(v_1,v_2) = (v_2,v_2)\)
    • Homework: 3, 4, 6, 7, 8, 10, 12, 13, 14, 17
    • Hand In: 8**, 14*:
      • 8**. Find the range and kernel (like the column space and nullspace) of \(T\):
        • (a) \(T(v_1,v_2) = (v_1-2v_2,0)\)
        • (b) \(T(v_1,v_2,v_3) = (v_3,v_1)\)
        • (c) \(T(v_1,v_2,v_3) = (0,v_1,0)\)
        • (d) \(T(v_1,v_2,v_3) = (v_1,v_1+v_2,v_2)\)
      • 14*. Follow the directions for Problem 14 in Problem Set 8.1, with \[A = \left[\matrix{4&3\cr 3&2}\right].\]
  • Section 8.2
    • In class: 10*:
      • 10*. Follow the directions for Problem 10 in Problem Set 8.2, with \[T({\bf v}_1) = {\bf w}_1+{\bf w}_2,\quad T({\bf v}_2) = {\bf w}_2+ {\bf w}_3,\quad T({\bf v}_3) = {\bf w}_3.\]
    • Homework: 1, 5, 6, 8, 10, 11, 12, 14, 15, 17, 18, 20, 30
    • Hand In: 10**, 15*, 30*:
      • 10**. Follow the directions for Problem 10 in Problem Set 8.2, with \[T({\bf v}_1) = {\bf w}_1+{\bf w}_3,\quad T({\bf v}_2) = {\bf w}_1 + {\bf w}_2 + {\bf w}_3,\quad T({\bf v}_3) = {\bf w}_1.\]
      • 15*. Follow the directions for Problem 15 in Problem Set 8.2, with
        • (a) \(M\) transforming \((1,0)\) to \((a,b)\) and \((0,1)\) to \((c,d)\)
        • (b) \(N\) transforming \((r,s)\) to \((1,0)\) and \((t,u)\) to \((0,1)\)
        • Answer part (c) with this \(N\) (condition is on \(r, s, t, u\)).
      • 30*. Follow the directions for Problem 30 in Problem Set 8.2, but with \(S\) reflection across the \(x\)-axis.
  • Section 8.3
    • In class: 2*:
      • 2*. Follow the directions for Problem 2 in Problem Set 8.3, with \(A_1 = \left[\matrix{0&3\cr 0&0}\right]\) and \(A_2 = \left[\matrix{-2&2\cr -2&2}\right]\). (For each part solve a system found by computing \(A_iB_i\) and \(B_iJ\) for \(B_i = \left[\matrix{a&b\cr c&d}\right]\), then setting the two products equal. Your matrices \(B_i\) will not be unique.)
    • Homework: 2, 3, 4
    • Hand In: 2**:
      • 2**. Follow the directions for Problem 2 in Problem Set 8.3, with \(A_1 = \left[\matrix{0&5\cr 0&0}\right]\) and \(A_2 = \left[\matrix{-3&9\cr -1&3}\right]\). (For each part solve a system found by computing \(A_iB_i\) and \(B_iJ\) for \(B_i = \left[\matrix{a&b\cr c&d}\right]\), then setting the two products equal. Your matrices \(B_i\) will not be unique.)