Fall 2024

Math 3110 Linear Algebra for Engineers

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Homework 6

Due Wednesday October 9, in class
  • Section 3.5
    • Read Section 3.5.
    • Watch Lecture 10: The Four Fundamantal Subspaces and Lecture 11: Matrix spaces; rank 1; small world graphs of the MIT OpenCourseWare series
    • Worked example (group work): 3*.
      • 3*. Follow the directions for Problem 3 in Problem Set 3.5, with \[A = \left[\matrix{1&2&3&4&5\cr -1&-2&-2&-2&-2\cr 0&0&-1&-2&-3}\right] = \left[\matrix{1&0&0\cr -1&1&0\cr 0&-1&1}\right]\left[\matrix{1&2&3&4&5\cr 0&0&1&2&3\cr 0&0&0&0&0}\right]\]
    • Homework: 1, 3, 4, 6, 7, 9-13 odd, 16, 17-21 odd, 23-27
    • Hand In: 3**, 17*, 25*:
      • 3**. Follow the directions for Problem 3 in Problem Set 3.5, with \[A = \left[\matrix{1&2&3&4&5\cr 1&2&4&6&8\cr 0&0&1&3&5\cr 0&0&0&1&2}\right] = \left[\matrix{1&0&0&0\cr 1&1&0&0\cr 0&1&1&0\cr 0&0&1&1}\right]\left[\matrix{1&2&3&4&5\cr 0&0&1&2&3\cr 0&0&0&1&2\cr 0&0&0&0&0}\right]\]
      • 17*. Follow the directions for Problem 17 in Problem Set 3.5, with \[A = \left[\matrix{0&0&1\cr 0&0&0\cr 0&0&0}\right]\] (Adjust the matrix \(I+A\) appropriately)
      • 25*. Follow the directions for Problem 25 in Problem Set 3.5, with
        • (a) \(A\) and \(A^t\) have the same number of non-pivots.
        • (b) \(A\) and \(A^t\) have the same nullspace.
        • (c) \(A\) may have the same row and column spaces.
  • Section 4.1
    • Read Section 4.1.
    • Watch Lecture 14: Orthogonal vectors and subspaces of the MIT OpenCourseWare series
    • Worked example (group work): 11*.
      • 11*. Draw Figure 4.1 to show each subspace correctly (give a basis for each subspace) for \[A = \left[\matrix{1&2&3\cr 3&6&9}\right]\ \mbox{and}\ B = \left[\matrix{1&0\cr 3&0\cr -2&0}\right]\].
    • Homework: 3, 5, 6, 9, 10, 11, 12, 14, 17, 18, 21, 24, 25, 28
    • Hand In: 11**, 17*:
      • 11**. Draw Figure 4.1 to show each subspace correctly (give a basis for each subspace) for \[A = \left[\matrix{1&2&3\cr 3&6&9\cr -1&-1&-1}\right]\ \mbox{and}\ B = \left[\matrix{1&0&0\cr 3&0&0\cr -2&0&0}\right]\].
      • 17*. Follow the directions for Problem 17 in Problem Set 4.1, with first the vector \((1,-1,1)\) and then the vectors \((1,-1,1)\) and \((1,1,0)\).