Spring 2018

Math 3110 Linear Algebra for Engineers

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Homework 2

Due Friday February 9, in class
  • Solution Manual: Chapter 2; Chapter 3
  • Section 2.3
    • Watch Lecture 2: Elimination with matrices of the MIT OpenCourseWare series
    • In class: 16*:
      • 16*. Follow the directions for Problem 16 in Problem Set 2.3, with the following parts:
        • (a) \(X\) is three times as old as \(Y\) and their ages add to \(36\).
        • (b) Use the two points \((2,4)\) and \((4,10)\).
    • Homework: 1, 3-5, 7, 8, 11, 13, 14, 16, 18, 22, 25, 26, 28, 29
    • Hand In: 4* (include solution to #3), 16**, 26*:
      • 4*. Follow the directions for Problem 4 in Problem Set 2.3, but use \({\bf b} = (1,1,0)\)
      • 16**. Follow the directions for Problem 16 in Problem Set 2.3, with the following parts:
        • (a) \(X\) is one and a half times as old as \(Y\) and their ages add to \(45\).
        • (b) Use the two points \((2,2)\) and \((4,-2)\).
      • 26*. Follow the directions for Problem 26 in Problem Set 2.3, but use \(A = \left[\matrix{2& 5\cr 1&3}\right]\).
  • Section 2.4
    • Watch Lecture 3: Multiplication and inverse matrices of the MIT OpenCourseWare series
    • In class: 21*, 32:
      • 21*. Follow the directions for Problem 21 in Problem Set 2.4, with the matrix \[A = \left[\matrix{0&-1&0&0\cr 0&0&-1&0\cr 0&0&0&-1\cr 0&0&0&0}\right]\]
    • Homework: 1, 2, 4, 6, 7, 9, 10, 11, 14, 15, 17, 18, 19, 20, 22, 26, 29, 32, 35
    • Hand In: 10*, 21**, 29*:
      • 10*. Let \[A = \left[\matrix{a&b&c\cr d&e&f\cr g&h&k}\right].\] Row 1 of \(A\) is added to row 3 to produce \(EA\). Then \(F\) adds row 2 of \(EA\) to row 1 of \(EA\), the result is \(F(EA)\)
        • (a) Compute \(EA\) and then \(F(EA)\) (you first need to decide what the matrices \(E\) and \(F\) are).
        • (b) Do those steps in the opposite order, that is, first add row 2 of \(A\) to row 1 by computing \(FA\). Then add row 1 of \(FA\) to row 3 of \(FA\) by computing \(E(FA)\).
        • (c) What law is not obeyed by matrix multiplication?
      • 21**. Follow the directions for Problem 21 in Problem Set 2.4, with the matrix \[A = \left[\matrix{0&-2&0&0\cr 0&0&-2&0\cr 0&0&0&-2\cr 0&0&0&0}\right]\]
      • 29*. Follow the directions for Problem 29 in Problem Set 2.4, with the matrix \[A = \left[\matrix{2&-2&3\cr 4&2&-1\cr -6&2&5}\right]\]
  • Section 2.5
    • In class: 25*:
      • 25*. Follow the directions for Problem 25 in Problem Set 2.5, with the matrices \[A = \left[\matrix{3&1&1\cr 1&3&1\cr 1&1&3}\right]\ \mbox{and}\ B = \left[\matrix{3&-1&-1\cr -1&3&-1\cr -1&-1&3}\right]\]
    • Homework: 1, 3, 5, 6, 7, 11, 12, 17, 19, 21, 23, 24, 25, 27, 30, 32, 35, 36
    • Hand In: 17*, 27*, 30*:
      • 17*.
        • (a) What \(3\) by \(3\) matrix \(E\) has the same effect as these steps? Subtract \(2\) times row 1 from row 2, subtract \(3\) times row 1 from row 3, then subtract row 2 from row 3.
        • (b) What single matrix \(L\) has the same effect as these three reverse steps? Add row 2 to row 3, add \(3\) times row 1 to row 3, then add \(2\) times row 1 to row 2.
      • 27*. Follow the directions for Problem 27 in Problem Set 2.5, with the matrices \[A = \left[\matrix{1&0&0\cr 3&1&5\cr 0&0&1}\right]\ \mbox{and}\ A = \left[\matrix{1&1&1\cr 1&3&3\cr 1&3&5}\right]\]
      • 30*. Follow the directions for Problem 30 in Problem Set 2.5, with the matrices \[A = \left[\matrix{a&-b&b\cr -a&a&-b\cr a&-a&a}\right]\ \mbox{and}\ C = \left[\matrix{3&c&c\cr c&c&c\cr 9&5&c}\right]\]
  • Section 2.6
    • Watch Lecture 4: Factorization into A = LU of the MIT OpenCourseWare series
    • In class: 13*:
      • 13*. Follow the directions for Problem 13 in Problem Set 2.6, with the matrix \[A = \left[\matrix{a&c&d&b\cr a&b&b&b\cr a&b&c&c\cr a&b&c&d}\right]\]
    • Homework: 1, 3, 4, 6, 7-9, 11, 12, 14, 15, 16, 19, 23
    • Hand In: 7*, 13**, 16*:
      • 7*. Follow the directions for Problem 7 in Problem Set 2.6, with the matrix \[A = \left[\matrix{1&1&-4\cr 3&2&-10\cr -1&1&3}\right]\]
      • 13**. Follow the directions for Problem 13 in Problem Set 2.6, with the matrix \[A = \left[\matrix{a&d&b&c\cr a&b&b&b\cr a&b&c&c\cr a&b&c&d}\right]\]
      • 16*. Follow the directions for Problem 16 in Problem Set 2.6, with the matrices \[L = \left[\matrix{1&0&0\cr 3&1&0\cr -1&-2&1}\right]\ \mbox{and}\ U = \left[\matrix{1&1&-4\cr 0&-1&2\cr 0&0&3}\right]\ \mbox{and the vector}\ {\bf b} = \left[\matrix{8\cr 21\cr -8}\right]\]
  • Section 2.7
    • Watch Lecture 5: Transposes, permutations, spaces \({\mathbb{R}}^n\) of the MIT OpenCourseWare series
    • In class: 11*:
      • 11*. Follow the directions for Problem 11 in Problem Set 2.7, with the matrix \[A = \left[\matrix{0&-2&1\cr 0&0&3\cr 1&-1&2}\right]\]
    • Homework: 1, 2, 4, 7, 11, 15, 17, 18, 20, 21, 24, 26, 30, 32
    • Hand In: 11**, 21*, 26*:
      • 11**. Follow the directions for Problem 11 in Problem Set 2.7, with the matrix \[A = \left[\matrix{0&0&-2&1\cr 1&-1&2&3\cr 0&0&0&-2\cr 0&3&4&2}\right]\]
      • 21*. Follow the directions for Problem 21 in Problem Set 2.7, with the matrices \[S = \left[\matrix{1&3&-5\cr 3&2&2\cr -5&2&7}\right]\ \mbox{and}\ S = \left[\matrix{-1&b&c\cr b&d&e\cr c&e&f}\right]\]
      • 26*. Follow the directions for Problem 26 in Problem Set 2.7, with the matrices \[S = \left[\matrix{1&-2&0\cr -2&7&6\cr 0&6&10}\right]\ \mbox{going toward}\ D = \left[\matrix{1&0&0\cr 0&3&0\cr 0&0&-2}\right].\] (You'll need to use different multipliers, that is, in (a), remove the \(-2\) below the pivot; in (b), remove the \(6\) below the pivot)
  • Section 3.1
    • Watch Lecture 6: Column Space and Null Space of the MIT OpenCourseWare series
    • In Class: 10*:
      • 10*. Which of the following subsets of \(\mathbb{R}^3\) are actually subspaces of \(\mathbb{R}^3\)?
        • (a) The plane of vectors \((b_1,b_2,b_3)\) with \(b_3 = b_1+b_2\).
        • (b) The plane of vectors \((b_1,b_2,b_3)\) with \(b_2 = -1\).
        • (c) The vectors \((b_1,b_2,b_3)\) with \(b_1b_2=0\).
        • (d) All linear combinations of \({\bf v} = (-1,2,3)\) and \({\bf w} = (0,1,4)\).
        • (e) All vectors \((b_1,b_2,b_3)\) satisfying \(b_1+b_2+b_3=3\).
        • (f) All vectors \((b_1,b_2,b_3)\) with \(b_1 \leq b_3\).
    • Homework: 1, 3, 7, 10, 11, 15, 16, 17, 19, 20, 22, 23, 25, 28
    • Hand In: 10**, 22*, 28*:
      • 10**. Which of the following subsets of \(\mathbb{R}^3\) are actually subspaces of \(\mathbb{R}^3\)?
        • (a) The plane of vectors \((b_1,b_2,b_3)\) with \(b_3 = b_1+1\).
        • (b) The plane of vectors \((b_1,b_2,b_3)\) with \(b_3 = 0\).
        • (c) The vectors \((b_1,b_2,b_3)\) with \(b_2b_3=1\).
        • (d) All linear combinations of \({\bf v} = (-1,-1,2)\) and \({\bf w} = (-2,2,-4)\).
        • (e) All vectors \((b_1,b_2,b_3)\) satisfying \(b_1+2b_2+3b_3=0\).
        • (f) All vectors \((b_1,b_2,b_3)\) with \(b_1 \geq b_3\).
      • 22*. Follow the directions for Problem 22 in Problem Set 3.1, with the matrices \[ \left[\matrix{1&0&1\cr 0&1&0\cr 0&0&1}\right], \left[\matrix{1&0&1\cr 0&1&0\cr 0&0&0}\right]\ \mbox{and}\ \left[\matrix{1&1&1\cr 0&1&1\cr 0&1&1}\right]\]
      • 28*. Follow the directions for Problem 28 in Problem Set 3.1, with the column space instead containing \((-1,2,-3)\) and \((1,1,-1)\) but not \((0,3,1)\). For the second part make the column space be the line containing the vector \((1,1,-1)\).