Spring 2023

Math 3110 Linear Algebra for Engineers

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Homework 6

Due Friday April 21, in class
  • Solution Manual: Chapter 6; Chapter 7.
  • Section 6.2
    • Read Section 6.2. WeBWorK reading question due April 5 @ 8:50 am.
    • Watch Lecture 22: Diagonalization and Powers of A of the MIT OpenCourseWare series.
    • In class: 2*. Final due date: April 19.
      • 2*. Follow the directions for Problem 2 in Problem Set 6.2, with \(\lambda_1 = 3, {\bf x}_1 = \left[\matrix{0\cr 1}\right], \lambda_2 = -2\ \mbox{and}\ {\bf x}_2 = \left[\matrix{-1\cr 1}\right].\)
    • Homework: 1, 2, 4, 6, 11, 12, 15-18, 21, 25, 26, 27, 34
    • Hand In: 2**, 16*, 18*:
      • 2**. Follow the directions for Problem 2 in Problem Set 6.2, with \(\lambda_1 = -1, {\bf x}_1 = \left[\matrix{3\cr 1}\right], \lambda_2 = 2\quad \mbox{and}\quad {\bf x}_2 = \left[\matrix{5\cr 2}\right].\)
      • 16*. Follow the directions for Problem 16 in Problem Set 6.2, with \( A_1 = \left[\matrix{0.7&0.8\cr 0.3&0.2}\right]\).
      • 18*. Diagonalize \(A = \left[\matrix{3&-2\cr -2&3}\right]\) and compute \(X\Lambda^kX^{-1}\) to find a formula for \(A^k\).
  • Section 6.4
    • Read Section 6.4. WeBWorK reading question due April 12 @ 8:50 am.
    • Watch Lecture 25: Symmetric matrices and positive definiteness of the MIT OpenCourseWare series.
    • In class: 8*. Final due date: April 21.
      • 8*. Follow the directions for Problem 8 in Problem Set 6.4, with \(S = \left[\matrix{4&6\cr 6&9}\right].\)
    • Homework: 1, 3, 5, 7, 8, 13, 15, 16, 19, 21, 23, 28
    • Hand In: 8**, 28*:
      • 8**. Follow the directions for Problem 8 in Problem Set 6.4, with \(S = \left[\matrix{9&15\cr 15&25}\right].\)
      • 28*: Follow the directions for Problem 28 in Problem Set 6.4, with \(A = \left[\matrix{1&10^{-12}\cr 0&1-10^{-12}}\right].\)
  • Section 6.5
    • Read Section 6.5. WeBWorK reading question due April 14 @ 8:50 am.
    • Watch Lecture 27: Positive definite matrices and minima of the MIT OpenCourseWare series.
    • In class: 7*. Final due date: April 24.
      • 7*. Follow the directions for Problem 7 in Problem Set 6.5, with \[A = \left[\matrix{0&1\cr -1&2}\right], A = \left[\matrix{1&1\cr -1&0\cr 1&1}\right]\quad \mbox{and}\quad A = \left[\matrix{1&1&-2\cr 0&2&-2}\right].\]
    • Homework: 2, 3, 4, 7, 8, 11, 12, 14, 17, 20, 21, 22, 25
    • Hand In: 7**, 22*:
      • 7**. Follow the directions for Problem 7 in Problem Set 6.5, with \[A = \left[\matrix{2&-1\cr 3&2}\right], A = \left[\matrix{-1&-1\cr 2&2\cr 3&1}\right]\ \mbox{and}\ A = \left[\matrix{1&2&-4\cr -1&3&5}\right].\]
      • 22*. Follow the directions for Problem 22 in Problem Set 6.5, with \[S = \left[\matrix{13&12\cr 12&13}\right]\ \mbox{and}\ S = \left[\matrix{25&9\cr 9&25}\right].\]
  • Section 7.1
    • Read Section 7.1. WeBWorK reading question due April 17 @ 8:50 am.
    • Maple worksheet on SVD image compression of flags, Maple worksheet on SVD image compression of other images (the Maple code inputs images in tiff format). (Maple worksheet credit: John May (modified by Russell Blyth))
    • Watch Lecture 29: Singular Value Decomposition of the MIT OpenCourseWare series.
    • In class: 1*. Final due date: April 28.
      • 1*. What are the ranks \(r\) of the following matrices? Write \(A\) and \(B\) as the sum of \(r\) pieces \({\bf uv}^t\) of rank one. We do not require orthogonality of the different \({\bf u}_i\) and \({\bf v}_j\). \[A = \left[\matrix{-1&0&1\cr -2&0&2\cr 3&0&-3}\right], \ \mbox{and}\ B = \left[\matrix{1&0&1&-1\cr 0&1&2&-2\cr 1&1&3&-3\cr 2&2&6&-6}\right].\]
    • Homework: 1-4, 6, 8
    • Hand In: 1**, 3*:
      • 1**. What are the ranks \(r\) of the following matrices? Write \(A\) and \(B\) as the sum of \(r\) pieces \({\bf uv}^t\) of rank one. We do not require orthogonality of the different \({\bf u}_i\) and \({\bf v}_j\). \[A = \left[\matrix{1&2&-2&-1\cr 3&6&-6&-3\cr 2&4&-4&-2\cr 4&8&-8&-4}\right], \ \mbox{and}\ B = \left[\matrix{1&2&3&4\cr 2&4&9&16\cr 3&6&12&20\cr 1&2&6&12}\right].\]
      • 3*. Follow the directions for Problem 3 in Problem Set 7.1, with \[A_1 = \left[\matrix{2&2&1&1\cr 2&2&1&1\cr 1&1&1&1}\right], \ \mbox{and}\ A_2 = \left[\matrix{1&2&2&2\cr 1&1&1&1\cr 1&3&3&3}\right].\]
  • Section 7.2
    • Read Section 7.2. WeBWorK reading question due April 19 @ 8:50 am.
    • In class: 4*. Final due date: May 1.
      • 4*. Follow the directions for Problem 4 in Problem Set 7.2, with \[A = \left[\matrix{2&0&1\cr 2&1&0}\right].\]
    • Homework: 1-4, 8, 11, 13, 14, 17
    • Hand In: 4**, 8*:
      • 4**. Follow the directions for Problem 4 in Problem Set 7.2, with \[A = \left[\matrix{1&0&2\cr 1&2&0}\right].\]
      • 8*. Follow the directions for Problem 8 in Problem Set 7.2, with \(A = \left[\matrix{1&4\cr 2&8}\right]\) (compute the correct matrices \(A^tA\) and \(AA^t\)).