Spring 2023

Math 3110 Linear Algebra for Engineers

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Homework 5

Due Wednesday April 5, in class
  • Solution Manual: Chapter 4, Chapter 5, Chapter 6
  • Section 4.4
    • Read Section 4.4. WeBWorK reading question due March 20 @ 8:50 am.
    • Watch Lecture 17: Orthogonal matrices and Gram Schmidt of the MIT OpenCourseWare series.
    • In class: 5*. Final due date: April 3.
      • 5*. Follow the directions for Problem 5 in Problem Set 4.4, with the plane \(x+2y+2z=0\).
    • Homework: 1-3, 5, 6, 8-11, 13, 15, 18, 20, 21, 22, 30, 31
    • Hand In: 5**, 15*, 31*:
      • 5**. Follow the directions for Problem 5 in Problem Set 4.4, with the plane \(x+2y+3z=0\).
      • 15*. Follow the directions for Problem 15 in Problem Set 4.4, with the matrix \(A = \left[\matrix{-2&3\cr 1&-1\cr 2&-1}\right]\) and (in part (c)) the vector \((2,4,-2)\).
      • 31*. Follow the directions for Problem 31 in Problem Set 4.4, with the matrix \[\left[\matrix{1&1&1&1&1&1&1&1\cr 1&-1&1&-1&1&-1&1&-1\cr 1&1&-1&-1&1&1&-1&-1\cr 1&-1&-1&1&1&-1&-1&1\cr 1&1&1&1&-1&-1&-1&-1\cr 1&-1&1&-1&-1&1&-1&1\cr 1&1&-1&-1&-1&-1&1&1\cr 1&-1&-1&1&-1&1&1&-1}\right].\] Use the vector \({\bf b} = (1,1,1,-1,-1,-1,1,-1)\) for the second part.
  • Section 5.1
    • Read Section 5.1. WeBWorK reading question due March 27 @ 8:50 am.
    • Watch Lecture 18: Properties of determinants of the MIT OpenCourseWare series.
    • In class: 18*. Final due date: April 5.
      • 18*. Follow the directions for Problem 18 in Problem Set 5.1, with the matrix \[\left[\matrix{1&1&1\cr a&b&c\cr a^2&b^2&c^2}\right].\]
    • Homework: 1, 3, 4, 5, 7, 8, 11, 14, 15, 18, 19, 22, 23, 24, 27, 28
    • Hand In: 18**, 22*:
      • 18**. Follow the directions for Problem 18 in Problem Set 5.1, with the matrix \[\left[\matrix{1&a&a^2&a^3\cr 1&b&b^2&b^3\cr 1&c&c^2&c^3\cr 1&d&d^2&d^3}\right].\]
      • 22*. Follow the directions for Problem 22 in Problem Set 5.1, with the matrix \[A = \left[\matrix{3&1\cr 1&3}\right].\] (Compute with the new correct matrices \(A^{-1}\) and \(A-\lambda I\)).
  • Section 5.2
    • Read Section 5.2. WeBWorK reading question due March 29 @ 8:50 am.
    • Watch Lecture 19: Determinant formulas and cofactors of the MIT OpenCourseWare series.
    • In class: 2*. Final due date: April 12.
      • 2*. Follow the directions for Problem 2 in Problem Set 5.2, with the matrices \[A = \left[\matrix{1&0&1\cr 0&1&1\cr 1&1&0}\right]\ \mbox{and}\ B = \left[\matrix{2&3&4\cr 3&4&5\cr 4&5&6}\right].\]
    • Homework: 1-3, 11, 12, 13, 16, 23, 25
    • Hand In: 2**, 12*:
      • 2**. Follow the directions for Problem 2 in Problem Set 5.2, with the matrices \[A = \left[\matrix{1&1&1&0\cr 1&1&0&1\cr 1&0&1&1\cr 0&1&1&1}\right]\ \mbox{and}\ B = \left[\matrix{1&2&3&4\cr 5&6&7&8\cr 9&0&1&2\cr 3&4&5&6}\right].\]
      • 12*. Follow the directions for Problem 12 in Problem Set 5.2, with the matrix \[A = \left[\matrix{2&1&0\cr 1&2&1\cr 0&1&2}\right],\ \mbox{which has inverse}\ A^{-1} = \frac{1}{4}\left[\matrix{3&-2&1\cr -2&4&-2\cr 1&-2&3}\right].\]
  • Section 5.3
    • Read Section 5.3. WeBWorK reading question due March 31 @ 8:50 am.
    • Watch Lecture 20: Cramer's Rule, inverse matrix, and volume of the MIT OpenCourseWare series.
    • In class: 8*. Final due date: April 14.
      • 8*. Follow the directions for Problem 8 in Problem Set 5.3, with the matrix \[A = \left[\matrix{2&1&0\cr -1&2&-1\cr 0&1&2}\right].\] The matrix \(C\) has form \[C = \left[\matrix{5&2&-1\cr \cdot&\cdot&\cdot\cr \cdot&\cdot&\cdot}\right].\] Ignore the part about changing the value of one of the entries for this version of the problem.
    • Homework: 1, 2, 6, 8, 10, 11, 15, 17
    • Hand In: 1*, 8**:
      • 1*. Follow the directions for Problem 1 in Problem Set 5.3, with the systems \[\mbox{(a)}\ \matrix{3x_1&+&7x_2&=&5\cr x_1&+&3x_2&=&3}\] and \[\mbox{(b)}\ \matrix{2x_1&-&x_2& & &=&0\cr -x_1&+&2x_2&-&x_3&=&1\cr & &-x_2&+&2x_3&=&0}.\]
      • 8**. Follow the directions for Problem 8 in Problem Set 5.3, with the matrix \[A = \left[\matrix{1&-2&3\cr 2&3&-3\cr 3&-4&5}\right].\] Ignore the part about changing the value of one of the entries for this version of the problem. (This time there's no hint about the matrix \(C\).)
  • Section 6.1
    • Read Section 6.1. WeBWorK reading question due April 3 @ 8:50 am.
    • Watch Lecture 21: Eigenvalues and Eigenvectors of the MIT OpenCourseWare series.
    • Geogebra activity on eigenvectors (or another Geogebra activity on eigenvectors)
    • In class: 24*. Final due date: April 17.
      • 24*. Follow the directions for Problem 24 in Problem Set 6.1, with the matrix \[A = \left[\matrix{2\cr 1\cr 2}\right]\left[\matrix{1&2&1}\right]=\left[\matrix{2&4&2\cr 1&2&1\cr 2&4&2}\right].\]
    • Homework: 1, 2, 4, 6, 7, 9, 12, 13, 15, 17, 19, 24, 27, 28, 29, 32
    • Hand In: 2*, 24**, 27*:
      • 2*. Follow the directions for Problem 2 in Problem Set 6.1, with the matrices \[A = \left[\matrix{4&1\cr 3&2}\right]\quad \mbox{and}\quad A-I = \left[\matrix{3&1\cr 3&1}\right].\] Don't forget to complete the final sentence: "\(A - I\) has the ***** eigenvectors as \(A\). Its eigenvalues are ***** by \(1\)".
      • 24**. Follow the directions for Problem 24 in Problem Set 6.1, with the matrix \[A = \left[\matrix{1\cr 3\cr 1}\right]\left[\matrix{3&1&3}\right]=\left[\matrix{3&1&3\cr 9&3&9\cr 3&1&3}\right].\]
      • 27*. Follow the directions for Problem 27 in Problem Set 6.1, with the matrices \[A = \left[\matrix{1&1&1&0\cr 1&0&1&1\cr 1&1&1&0\cr 0&1&1&1}\right]\quad \mbox{and}\quad C = \left[\matrix{0&1&0&0\cr 1&0&0&0\cr 0&0&0&1\cr 0&0&1&0}\right].\]