# ME205: Numerical Methods for Engineers

Unit 4: Applications of Numerical Methods in Linear Algebra   *A witticism common among practitioners of mathematics is that a mathematical problem, to be solvable, must at some point be mapped onto a problem in linear algebra.  While not reflecting the entire truth, this viewpoint is sufficiently correct that, in this course, numerical methods and their application to engineering problems may often be understood in terms of fundamental problems in linear algebra.

This unit will focus on a few of those fundamental problems. After providing a review of some basic properties of matrices, matrix algebra, and their relationship to linear algebra, the unit then covers introductions to  solving systems of linear equations by elimination and iterative methods, techniques for finding the inverse of a matrix, and the concept of singular value decomposition for non-square matrices. *

Time Advisory: This unit will take you approximately 18 hours to complete.

☐    Subunit 4.1: 3 hours

☐    Subunit 4.2: 3 hours

☐    Subunit 4.3: 3 hours

☐    Subunit 4.4: 3 hours

☐    Subunit 4.5: 3 hours

☐    Subunit 4.6: 3 hours

Unit4 Learning Outcomes
Upon successful completion of this unit, the student will be able to:

• Define and identify special types of matrices.
• Perform basic matrix operations.
• Define and perform Gaussian elimination to solve a linear system.
• Identify pitfalls of Gaussian elimination.
• Define and perform Gauss-Seidel method for solving a linear system.
• Use LU decomposition to find the inverse of a matrix.
• Define and perform singular value decomposition; explain the significance of singular value decomposition.

4.1 Overview of Linear Algebra   - Reading: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapters on Matrix Algebra” Link: University of South Florida: Holistic Numerical Methods Institute’s Textbook Chapters on Matrix Algebra: Introduction (PDF), Vectors (PDF), Binary Matrix Operations (PDF), Unary Matrix Operations (PDF), and Systems of Equations (PDF)

Instructions: Please read all five textbook chapters (Introduction, Vectors, Binary Matrix Operations, Unary Matrix Operations, and Systems of Equations) in their entirety.

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Reading: Stanford University: Professor Stephen Boyd’s “A Primer on Matrices”

Link: Stanford University: Professor Stephen Boyd’s “A Primer on Matrices” (PDF)

Instructions: Please read and work through the examples before watching Professor Strang’s lecture on Key Ideas of Linear Algebra in this subunit.

• Lecture: MIT Opencourseware: Professor Gilbert Strang’s Mathematics 18.085: “Recitation 1: Key Ideas of Linear Algebra” Link: MIT Opencourseware: Professor Gilbert Strang’s Mathematics 18.085: “Recitation 1: Key Ideas of Linear Algebra” (FLASH, MP4, or iTunes)

Also available in:

Instructions: Please view the video lecture in its entirety (49:31 minutes).

• Assessment: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Background of Simultaneous Linear Equations” Link: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Introduction of Background of Simultaneous Linear Equations” (PDF)

Instructions: Please attempt all questions for this multiple choice quiz.  You can find the answers here (PDF).

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

4.2 Gaussian Elimination   - Web Media: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “Gaussian Elimination” Link: University of South Florida: Holistic Numerical Methods Institute’s "Lectures on “Gaussian Elimination” (YouTube)

Instructions: Under “Digital Audiovisual Lectures,” please select the YouTube link after the title of each lecture.  View all 17 lectures on Gaussian Elimination in their entirety (approximately 2.25 hours).

• Reading: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter on Gaussian Elimination” Link: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter on Gaussian Elimination” (PDF)

Instructions: Read the entire chapter (27 pages).  Based on your reading about numerical differentiation, can you think of ways to improve Gaussian Elimination?

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Assessment: University of South Florida: Holistic Numerical Methods Institute’s Test Your Knowledge of Gaussian Elimination” Link: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Gaussian Elimination” (PDF)

Instructions: Please attempt all questions on this multiple choice quiz.  You can find the answers here (PDF).

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Reading: University of South Florida: Holistic Numerical Methods Institute’s “Mechanical Engineering Example of Gaussian Elimination” Link: University of South Florida: Holistic Numerical Methods Institute’s “Mechanical Engineering Example of Gaussian Elimination” (PDF)

Instructions: Follow the entire example, and try to reproduce the calculations as you read.

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

4.3 Gauss-Seidel Method   - Web Media: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “Gauss-Seidel Method” Link: University of South Florida: Holistic Numerical Methods Institute’s "Lectures on “Gauss-Seidel Method” (YouTube)

Instructions: Please view the six video lectures in their entirety.  To access the lectures, select the YouTube link after each title under the heading “Digital Audiovisual Lectures.”  The total run time is approximately is 38 minutes.

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• Reading: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter on Gauss-Seidel Method” Link: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter on Gauss-Seidel Method” (PDF)

Instructions: Read the entire chapter (15 pages).  Based on your reading about numerical differentiation, can you think of ways to improve Gauss-Seidel method?

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Assessment: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Gauss-Seidel Method” Link: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Gauss-Seidel Method” (PDF)

Instructions: Please attempt all questions on the multiple choice quiz.  You can find the answers here (PDF).

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Reading: University of South Florida: Holistic Numerical Methods Institute’s “Mechanical Engineering Example of Gauss-Seidel Method” Link: University of South Florida: Holistic Numerical Methods Institute’s “Mechanical Engineering Example of Gauss-Seidel Method” (PDF)

Instructions: Read through the example in its entirety, and try to reproduce the calculations.

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

4.4 LU Decomposition   - Web Media: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “LU Decomposition” Link: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “LU Decomposition” (YouTube)

Instructions: View the 8 video lectures under the heading “Digital Audiovisual Lectures.”  To access the videos, click on the YouTube link after each video’s title.  The total run time is approximately 48 minutes.

• Reading: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter on LU Decomposition” Link: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter on LU Decomposition” (PDF)

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Assessment: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of LU Decomposition” Link: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of LU Decomposition” (PDF)

Instructions: Work through each question on the multiple choice quiz.  You can find the answers here (PDF).

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Reading: University of South Florida: Holistic Numerical Methods Institute’s “Chemical Engineering Example of LU Decomposition” Link: University of South Florida: Holistic Numerical Methods Institute’s “Chemical Engineering Example of LU Decomposition” (PDF)

Instructions: Follow the entire example, and try to reproduce the calculations as you read.

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Web Media: MIT Opencourseware: Professor Gilbert Strang’s Mathematics 18.06: “Factorization into A = LU”  Link: MIT Opencourseware: Professor Gilbert Strang’s Mathematics 18.06: “Factorization into A = LU” (FLASH, MP4, or iTunes)

Also available in:

Instructions: Please view the entire lecture (50:13 minutes) before moving on to the reading in this subunit.  You may also access the transcript of the video by clicking on the “transcript” tab and then the “download this transcript” link on the webpage.

4.5 Singular Value Decomposition   - Web Media: MIT Opencourseware: Professor Gilbert Strang’s Mathematics 18.06: “Singular Value Decomposition” Link: MIT Opencourseware: Professor Gilbert Strang’s Mathematics 18.06: “Singular Value Decomposition” (FLASH, MP4, or iTunes)

Also available in:

Instructions: Please view the entire lecture (41:34 minutes) before moving on to the reading in this subunit.  You may also access the transcript of the video by clicking on the “transcript” tab and then the “download this transcript” link on the webpage.