Fall 2024

Math 3110 Linear Algebra for Engineers

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Homework 5

Due Wednesday October 2, in class
  • Section 3.3
    • Read Section 3.3.
    • Watch Lecture 8: Solving \(A{\bf x}={\bf b}\): row reduced form \(R\) of the MIT OpenCourseWare series
    • Worked example: TBA 1*.
      • 1*. Follow the directions for Problem 1 in Problem Set 3.3, with \[A = \left[\matrix{1&-1&4&2\cr 2&-1&7&-1\cr 3&-1&10&-4}\right]\] and set \[{\bf b} = \left[\matrix{b_1\cr b_2\cr b_3}\right]\ \mbox{equal to}\ \left[\matrix{3\cr -1\cr -5}\right]\ \mbox{in Parts 5 and 6.}\]
    • Homework: 2, 1, 4, 6, 7, 8, 11, 13, 14, 16, 18, 20, 23, 25, 26
    • Hand In: 1**, 18*, 26*:
      • 1**. Follow the directions for Problem 1 in Problem Set 3.3, with \[A = \left[\matrix{2&3&-1&2\cr 4&8&-4&6\cr 6&17&-11&14}\right]\] and set \[{\bf b} = \left[\matrix{b_1\cr b_2\cr b_3}\right]\ \mbox{equal to}\ \left[\matrix{-1\cr -2\cr -3}\right]\ \mbox{in Parts 5 and 6.}\]
      • 18*. Follow the directions for Problem 18 in Problem Set 3.3, with \[A = \left[\matrix{2&-1&2&3\cr -4&4&-3&0}\right]\ \mbox{and}\ A = \left[\matrix{1&0&1&1\cr -3&2&-2&-4\cr 0&4&7&6}\right].\]
      • 26*. Follow the directions for Problem 26 in Problem Set 3.3, with \[U = \left[\matrix{2&0&8&0\cr 0&0&4&0\cr 0&0&0&0}\right]\ \mbox{and}\ {\bf c} = \left[\matrix{5\cr 2\cr 0}\right].\] (The system (three equations, four unknowns) is first augmented with a zero right hand side, and then with \({\bf c}\) as right hand side. The number of free variables will be 2, not 1.)
  • Section 3.4
    • Read Section 3.4.
    • Watch Lecture 9: Independence, Basis and Dimension of the MIT OpenCourseWare series
    • Worked example: 16*.
      • 16*. Find a basis for each of these subspaces of \(\mathbb{R}^4\):
        • (a) All vectors whose first three components are equal.
        • (b) All vectors whose first two components add to zero.
        • (c) All vectors that are perpendicular to \((1,0,0,1)\ \mbox{and}\ (1,1,1,0)\).
        • (d) The column space and the nullspace of the \(4 \times 4\) identity matrix \(I\).
    • Homework: 2, 1, 3-5, 7-12, 15, 16, 18, 19, 24, 26, 27, 32, 41
    • Hand In: 7*, 16**, 27:
      • 7*. Follow the directions for Problem 7 in Problem Set 3.4, with \({\bf v}_1 = {\bf w}_1+{\bf w}_2+{\bf w}_3, {\bf v}_2 = {\bf w}_1-{\bf w}_2, {\bf v}_3 = -2{\bf w}_2-{\bf w}_3\).
      • 16**. Find a basis for each of these subspaces of \(\mathbb{R}^4\):
        • (a) All vectors whose first two components are equal.
        • (b) All vectors whose first three components add to zero.
        • (c) All vectors that are perpendicular to \((1,0,1,0)\ \mbox{and}\ (1,1,0,1)\).
      • 27. As in the textbook.