Spring 2018

Math 3110 Linear Algebra for Engineers

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Homework 4

Due Friday March 9, in class
  • Solution Manual: Chapter 4
  • Section 4.1
    • Watch Lecture 14: Orthogonal vectors and subspaces of the MIT OpenCourseWare series
    • In class: 11*:
      • 11*. Draw Figure 4.2 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 4.2 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 \((1,1,0)\).
  • Section 4.2
    • Watch Lecture 15: Projections onto subspaces of the MIT OpenCourseWare series
    • In class: 5*:
      • 5*. Follow the directions for Problem 5 in Problem Set 4.2, with \({\bf a}_1 = (2,-1,2)\) and \({\bf a}_2 = (-1,2,2)\).
    • Homework: 1-3, 5, 6, 8, 9, 11, 13, 16-18, 20, 19, 21, 23
    • Hand In: 5**, 13*, 19*:
      • 5**. Follow the directions for Problem 5 in Problem Set 4.2, with \({\bf a}_1 = (1,-2,3)\) and \({\bf a}_2 = (-2,2,2)\).
      • 13*. Follow the directions for Problem 13 in Problem Set 4.2, but with the third column removed instead of the last column.
      • 19*. Follow the directions for Problem 19 in Problem Set 4.2, but with the plane \(2x+y-4z=0\).
  • Section 4.3
    • Watch Lecture 16: Projection matrices and least squares of the MIT OpenCourseWare series
    • In class: 6*:
      • 6*. Follow the directions for Problem 6 in Problem Set 4.3, with \({\bf b} = (-1,1,4,4)\).
    • Homework: 1-3, 6, 7, 9, 17, 21, 22
    • Hand In: 6**, 22*:
      • 6**. Follow the directions for Problem 6 in Problem Set 4.3, with \({\bf b} = (-1,1,4,3,8)\) and \({\bf a} = (1,1,1,1,1)\).
      • 22*. Follow the directions for Problem 22 in Problem Set 4.3, with \(b = 5, 2, -2, -2, -3\) (that is, fit a line to the points \((t,b)=(-2,5),(-1,2),(0,-2),(1,-2),(2,-3)\)).