A printed version of this page has no headers, footers, menu bars or this message.

Due Wednesday August 28, in class

• Section 1.1
  • Read the Preface (really, it's the start of the course) and Section 1.1.
  • Watch Lecture 1: The geometry of linear equations of the MIT OpenCourseWare series
  • A Geogebra linear combination activity
  • Worked example (group work): 11*.
    • 11*. Follow the directions for Problem 11 in Problem Set 1.1, with the three points \((2,1), (1,3)\ \mbox{and}\ (3,2)\). Draw all three parallelograms.
  • Homework: 1, 3, 5, 7, 8, 11, 12, 13, 14, 15, 17, 19, 20, 21, 24, 27
  • Hand In: 7*, 8*, 11**:
    • 7*. Complete Problem 7 in Problem Set 1.1, but with the vectors \[{\bf u} = \left[\matrix{2\cr -3\cr 1\cr}\right], {\bf v} = \left[\matrix{-2\cr 3\cr -1\cr}\right]\ \mbox{and}\ {\bf w} = \left[\matrix{2\cr -5\cr -1\cr}\right]\](Don't assume that \({\bf w}\) must be a linear combination of \({\bf u}\) and \({\bf v}\) in this version of the problem.)
    • 8*. Complete Problem 8 in Problem Set 1.1, but with the vectors \[{\bf v} = \left[\matrix{-2\cr 3\cr -1\cr}\right]\ \mbox{and}\ {\bf w} = \left[\matrix{2\cr -1\cr -1\cr}\right]\]
    • 11**. Follow the directions for Problem 11 in Problem Set 1.1, with the three points \((-1,1), (2,1)\ \mbox{and}\ (1,3)\). Draw all three parallelograms.
• Section 1.2
  • Read Section 1.2.
  • Worked example (group work): 7*.
    • 7*. Find the angle \(\theta\) (from its cosine) between these pairs of vectors:
      • (a) \({\bf v} = \left[\matrix{\sqrt{3}\cr -1\cr}\right]\) and \({\bf w} = \left[\matrix{1\cr 0\cr}\right]\)
      • (b) \({\bf v} = \left[\matrix{1\cr 2\cr 2\cr}\right]\) and \({\bf w} = \left[\matrix{-2\cr 1\cr -2\cr}\right]\)
      • (c) \({\bf v} = \left[\matrix{1\cr 3\cr}\right]\) and \({\bf w} = \left[\matrix{1\cr 2\cr}\right]\)
  • Homework: 1, 2, 4, 7, 9, 13, 14, 25
  • Hand In: 7**, 13*, 25*:
    • 7**. Find the angle \(\theta\) (from its cosine) between these pairs of vectors:
      • (a) \({\bf v} = \left[\matrix{-\sqrt{3}\cr -1\cr}\right]\) and \({\bf w} = \left[\matrix{1\cr 0\cr}\right]\)
      • (b) \({\bf v} = \left[\matrix{1\cr 2\cr -2\cr}\right]\) and \({\bf w} = \left[\matrix{-2\cr 1\cr 2\cr}\right]\)
      • (c) \({\bf v} = \left[\matrix{1\cr -3\cr}\right]\) and \({\bf w} = \left[\matrix{1\cr 2\cr}\right]\)
    • 13*. Find nonzero vectors \({\bf u}, {\bf v}\) and \({\bf w}\) that are perpendicular to \({\bf n} = \left[\matrix{2\cr 1\cr -1\cr -1}\right]\) and to each other. (Hint: First find a vector \({\bf u} = \left[\matrix{u_1\cr 0\cr 0\cr u_4}\right]\) perpendicular to \({\bf n}\). Then find a vector \({\bf v} = \left[\matrix{v_1\cr 0\cr v_3\cr v_4}\right]\) perpendicular to \({\bf u}\) and \({\bf n}\). Then find a vector \({\bf w}\) perpendicular to \({\bf u}, {\bf v}\) and \({\bf n}\))
    • 25*. Complete Problem 25 in Problem Set 1.2, but with \({||\bf v}||=4\) and \({||\bf w}||=7\)
• Section 1.3
  • Read Section 1.3.
  • Worked example (group work): 6*.
    • 23*. For each matrix determine which number(s) \(c\) give dependent columns and give a combination of columns that equals the zero vector:
      • (a) \(\left[\matrix{2&1&1\cr 4&2&2\cr 8&3&c}\right]\)
      • (b) \(\left[\matrix{0&1&c\cr 1&0&-1\cr 1&1&0}\right]\)
      • (c) \(\left[\matrix{c&c&c\cr 1&1&1\cr 2&3&1\cr}\right]\)
  • Homework: 1, 3, 4, 9, 14, 15, 20, 21, 23 (this is four different problems in one)
  • Hand In: 21*, 23**:
    • 21*. Follow the directions for Problem 21 in Problem Set 1.3, but with the matrix \[S = \left[\matrix{1&0&0\cr -1&1&0\cr 1&-1&1\cr}\right].\]
    • 23**. For each matrix determine which number(s) \(c\) give dependent columns and give a combination of columns that equals the zero vector:
      • (a) \(\left[\matrix{1&1&0\cr 4&1&3\cr 6&2&c}\right]\)
      • (b) \(\left[\matrix{1&1&0\cr 0&1&c\cr 1&0&-1}\right]\)
      • (c) \(\left[\matrix{c&c&c\cr 2&2&2\cr 1&3&-3\cr}\right]\)