MTH5212 - Applied Linear Algebra - 2023/24

Topic outline

• Module content
• IMPORTANT module information
• Assessment information
• Online reading list
• Q-Review
• Early feedback questionnaire

• Announcements

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    Module announcements Forum
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    Student forum

    This forum is available for everyone to post messages to. You are strongly encouraged to ask all your questions using this forum, which is checked regularly by the module leaders. Feel free to reply to other students if you are able to.

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    Zoom link and recordings External tool

    Recordings of the lectures can be found in this link under the "Cloud Recordings" tab. If a password is required, you should also be able to find it there.

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    Lecture notes File

    587.7 KB

    Follow this link for the typeset lecture notes for the whole module. The presentation of these topics in the lectures will also be based on the following book:

    • Gilbert Strang, Introduction to Linear Algebra, 6th edition.

    Note that previous editions of Strang's book also contain all the material we will cover, but sometimes grouped slightly differently into sections.
    
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    Handwritten notes from lectures Folder

    Archive of handwritten notes from the module lectures.

  • Recommended study resources

    There are thousands of linear algebra resources available, including many books, exercise lists, videos, etc. Here are a couple of recommendations:

    • Linear Algebra and its Applications by Lay, Lay, and McDonald (available online from the QMUL library)
      Great textbook for reviewing the theory, with many practice exercises.
    • Essence of linear algebra by 3Blue1Brown
      YouTube playlist with great explanations and visualisations for getting the intuition behind several linear algebra concepts.
    • Archive of past final exams
      A database of final exams at QMUL. Search for "Applied Linear Algebra".
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    Revision Lecture for Final Exam (Part 1) 🆕 QMplus Media Video
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    Revision Lecture for Final Exam (Part 2) 🆕 QMplus Media Video

• Week 1

  TOPICS

  1. Linear Equations: lecture notes 1.1, 2.3, 2.4; Strang's book 2.1.
     
  2. Gauss elimination: lecture notes 1.2, 2.5; Strang's book 2.1, 2.2.
  3. Matrix multiplication, matrix inverses, Gauss-Jordan inversion: lecture notes 2.1, 2.5, 2.6; Strang's book 2.2.

  TASKS 

  1. Attend all the lectures on Monday and Thursday, and also your seminar session on Monday. 
  2. Review the corresponding sections in the lecture notes (and/or Strang's book, if you have it).
  3. Ask any questions you have in the student forum. 
  4. Do Coursework 0 on WeBWorK.

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    Coursework 0 File

    A tutorial for how to set up and access your WeBWorK account. Does not count for your final mark. 

  • Some real-life applications of Linear Algebra

    Optional material illustrating applications of Linear Algebra:

    https://www.youtube.com/watch?v=rowWM-MijXU
    https://www.youtube.com/watch?v=i8FukKfMKCI
    https://www.youtube.com/watch?v=TBpDMLCC2uY
    https://www.youtube.com/watch?v=cvcAjgMUPUA
    

  • Monday lecture recording

    Unfortunately the first lecture on Monday Sep 25th at 10:00 ended up being recorded on the wrong Zoom link. You can still find the recording following the Zoom link above, under the title "MTH5212 - Applied Linear Algebra - 2023/24 - Monday Seminar 2".
    

• Week 2

  TOPICS

  1. Vector spaces, subspaces: lecture notes 4.1, 4.2; Strang's book 3.1.
     
  2. Column space, nullspace: lecture notes 4.2, 4.8; Strang's book 3.1, 3.2.
  3. Computing the nullspace (solving Ax=0): lecture notes 4.2; Strang's book 3.2.

  TASKS 

  1. Attend all the lectures on Monday and Thursday, and also your seminar session on Monday. 
  2. Review the corresponding sections in the lecture notes (and/or Strang's book, if you have it).
  3. Ask any questions you have in the student forum. 
  4. Do Coursework 1 on WeBWorK.

• Week 3

  TOPICS

  1. Determinants and their properties: lecture notes 3.1, 3.2; Strang's book 5.1.
     
  2. Computing determinants, cofactors: lecture notes 3.2, 3.3; Strang's book 5.2.
  3. Cramer's rule: lecture notes 3.3; Strang's book 5.2, 5.3.

  TASKS 

  1. Attend all the lectures on Monday and Thursday, and also your seminar session on Monday. 
  2. Review the corresponding sections in the lecture notes (and/or Strang's book, if you have it).
  3. Ask any questions you have in the student forum. 
  4. Do Coursework 1 on WeBWorK.

  • Monday 15:00 seminar recording

    Due to technical problems, it was impossible to access Zoom during the 15:00 seminar session on Monday Oct 9th. The QReview recording should be available, and the 17:00 seminar session was recorded both on Zoom and QReview.
    

• Week 4

  TOPICS

  1. Solving Ax=b: lecture notes 1.2; Strang's book 3.3.
     
  2. Span, independence, bases: lecture notes 4.3, 4.4, 4.5, 4.6; Strang's book 3.4.

  TASKS 

  1. Attend all the lectures on Monday and Thursday, and also your seminar session on Monday. 
  2. Review the corresponding sections in the lecture notes (and/or Strang's book, if you have it).
  3. Ask any questions you have in the student forum. 
  4. Do Coursework 1 on WeBWorK before 11:59am on Wed 18th.

  
  

• Week 5

  TOPICS

  1. Bases, dimension: lecture notes 4.6; Strang's book 3.4.
     
  2. Coordinates: lecture notes 4.7; Strang's book 3.4.
  3. Rank, nullity, dimension of the column space and the row space: lecture notes 4.8; Strang's book 3.5.

  TASKS 

  1. Attend all the lectures on Monday and Thursday, and also your seminar session on Monday. 
  2. Review the corresponding sections in the lecture notes (and/or Strang's book, if you have it).
  3. Ask any questions you have in the student forum. 
  4. Do Coursework 2 on WeBWorK before 11:59am on Wed 1st.

• Week 6

  TOPICS

  1. Linear transformations: lecture notes 5.1, 5.2; Strang's book 8.1.
  2. The matrix of a linear transformation: lecture notes 5.3; Strang's book 8.2.

  TASKS 

  1. Attend all the lectures on Monday and Thursday, and also your seminar session on Monday. 
  2. Review the corresponding sections in the lecture notes (and/or Strang's book, if you have it).
  3. Ask any questions you have in the student forum. 
  4. Do Coursework 2 on WeBWorK before 11:59am on Wed 1st.

• Week 7: Reading week

  Reading week — no lectures or seminars.

• Week 8

  TOPICS

  1. Image and kernel of a linear transformation: lecture notes 5.8; Strang's book 8.1.
  2. Coordinatisation of a vector space: lecture notes 5.9; Strang's book 8.2.
  3. Change of coordinates: lecture notes 5.10; Strang's book 8.2.

  TASKS 

  1. Attend all the lectures on Monday and Thursday, and also your seminar session on Monday. 
  2. Review the corresponding sections in the lecture notes (and/or Strang's book, if you have it).
  3. Ask any questions you have in the student forum. 
  4. Do Coursework 3 on WeBWorK before 11:59am on Wed 22nd.

• Week 9

  TOPICS

  1. Eigenvalues and eigenvectors: lecture notes 6.1; Strang's book 6.1.
  2. Diagonalisation: lecture notes 6.2; Strang's book 6.2.

  TASKS 

  1. Attend all the lectures on Monday and Thursday, and also your seminar session on Monday. 
  2. Review the corresponding sections in the lecture notes (and/or Strang's book, if you have it).
  3. Ask any questions you have in the student forum. 
  4. Do Coursework 3 on WeBWorK before 11:59am on Wed 22nd.

  
  

• Week 10

  TOPICS

  1. Diagonalisation: lecture notes 6.2; Strang's book 6.2.
  2. Orthogonality: lecture notes 7.1; Strang's book 4.1.
  3. Orthogonal complements: lecture notes 7.2; Strang's book 4.1.

  TASKS 

  1. Attend all the lectures on Monday and Thursday, and also your seminar session on Monday. 
  2. Review the corresponding sections in the lecture notes (and/or Strang's book, if you have it).
  3. Ask any questions you have in the student forum. 
  4. Do Coursework 4 on WeBWorK before 11:59am on Wed Dec 6th.

• Week 11

  TOPICS

  1. Orthogonal and orthonormal sets: lecture notes 7.3, 7.4; Strang's book 4.1.
  2. Orthogonal projections: lecture notes 7.5; Strang's book 4.2.
  3. Gram-Schmidt process: lecture notes 7.6; Strang's book 4.4.
  4. Least-squares approximations: lecture notes 7.7; Strang's book 4.3.

  TASKS 

  1. Attend all the lectures on Monday and Thursday, and also your seminar session on Monday. 
  2. Review the corresponding sections in the lecture notes (and/or Strang's book, if you have it).
  3. Ask any questions you have in the student forum. 
  4. Do Coursework 4 on WeBWorK before 11:59am on Wed Dec 6th.

• Week 12

  TOPICS

  1. Least-squares approximations and applications: lecture notes 7.7; Strang's book 4.3.
     
  2. Revision lecture for final exam (posted above).

  TASKS 

  1. Attend the lecture on Monday morning. 
  2. Review the corresponding sections in the lecture notes (and/or Strang's book, if you have it).
  3. Ask any questions you have in the student forum. 
  4. Do Coursework 5 on WeBWorK before 11:59am on Wed Dec 20th.
  5. Watch the Revision Lecture for Final Exam (Part 1) and Revision Lecture for Final Exam (Part 2) posted above to prepare for the final exam.

• Online Reading List

• Assessment Information

  Assessment pattern:  80% final exam + 20% coursework.

  Format and dates for the in-term assessments: 5 coursework submissions on WebWork, each worth 4% of the final mark. Submission deadlines will be roughly every 2 weeks, on 18/10, 01/11, 22/11, 06/12, and 20/12.

  Format of final exam: 

  • The exam will be an in-person closed-book exam to take place in the regular January examination period. No paper, notes, or calculators will be allowed.
  • About two thirds of the exam marks will consist of multiple-choice questions (to be answered in a special multiple-choice answer sheet that you will be given, by filling circles with a pencil). The remaining one third of the exam marks will consist of open-ended questions that you will answer on the usual booklet.
  • The content examined will be very similar to the content examined in the past in-person closed-book exams (from 2020 and previous years). You are encouraged to prepare using these exams, together with the book linked in the "Recommended Study Resources" section of the QMPlus website.
  • The exam is designed to be solved in 2 hours, although you will have a total of 3 hours to write your answers. SpLD accommodations will be handled separately.
  • There will be a revision lecture in Week 12 posted online on QMPlus.

  Link to past papers: Past final exams can be found on this link.

  Description of feedback: Your coursework submissions will be instantly marked through WebWork, and you will normally have several attempts to get the answer right based on this feedback. In addition, official coursework solutions will be provided. You will be able to ask questions in the forum, which will be used to guide the content of the seminar sessions on Mondays. Weekly learning support hours will be provided as well.

• Module Description

  This module covers concepts in linear algebra and its applications. The ideas for two- and three-dimensional space covered by the first year module Vectors and Matrices will be developed and extended in a more general setting with a view to applications in subsequent pure and applied mathematics, probability and statistics modules. There will be a strong geometric emphasis in the presentation of the material and the key concepts will be illustrated by examples from various branches of science and engineering.

• Syllabus

  Applied Linear Algebra (in addition to Calculus/Analysis) is one of the most important parts of a mathematics course. This module is a rigorous first module in linear algebra. The ideas introduced in Vectors & Matrices for two- and three-dimensional space will be developed and extended in a more general setting, with a view to applications in subsequent modules in pure and applied mathematics, probability, and statistics.

  We will cover the following topics, in approximately the order shown:

  1. Systems of linear equations, matrix algebra, determinants

  2. Vector spaces (over the real and complex numbers): Definition and examples. Subspaces. Spanning sets. Linear independence. Bases and dimension of a vector space. Coordinate vectors and change-of-basis matrices. 

  3. Vector spaces associated with matrices: Row and column spaces, rank, null space. Rank-nullity theorem. 
     

  4. Linear transformations: Definition and examples. Matrix representations of linear transformations (with respect to given bases); composition of linear transformations as matrix products. 
     

  5. Eigenvalues and eigenvectors: Eigenvalues, eigenvectors, and eigenspaces. The characteristic polynomial. Diagonalisation of matrices. Applications such as calculating powers of matrices and solving systems of linear differential equations. 

  6. Orthogonality: Scalar product, definition and properties. Orthogonal and orthonormal sets. Gram-Schmidt process. Orthogonal complement of a subspace. Orthogonal projections. Orthogonal matrices. Applications such as least-squares estimates.

• Module Aims and Learning Outcomes

  ACADEMIC CONTENT

  This module covers:

  • Systems of linear equations, matrix algebra and determinants.
  • Vector spaces and linear transformations.
  • Orthogonality and the Gram-Schmidt process.
  • Eigenvalues, eigenvectors, the characteristic polynomial, and diagonalisation.

  
  

  DISCIPLINARY SKILLS

  At the end of this module, students should be able to:

  • Solve linear systems and write solutions in vector form.
  • Calculate the product of two matrices; calculate the transpose of a matrix; calculate the determinant, eigenvalues and eigenvectors of a square matrix; determine whether a given matrix is invertible; calculate the inverse of an invertible matrix; use algebraic equations of matrices.
  • Determine whether or not a given subset of a vector space is a subspace; whether or not a given vector is in the subspace spanned by a set of vectors; and whether given vectors are linearly independent and/or form a basis for a vector space or subspace.
  • Find the coordinates of a vector with respect to a given ordered basis; calculate the transition matrix corresponding to a change of basis; calculate the rank of a matrix; determine bases for the row and column spaces of a matrix.
  • Verify whether a mapping between vector spaces is linear, and if so calculate the matrix of the mapping with respect to given bases.
  • Calculate the scalar product of two vectors and determine whether the vectors are orthogonal and/or orthonormal; find the orthogonal projection of a vector onto a given subspace and the closest vector in a given subspace to a given vector.
  • Determine the set of least-squares solutions of a given linear system.
  • Apply the Gram-Schmidt process.
  • Apply standard results about diagonalisation of matrices as follows: for a real square matrix A with distinct eigenvalues, find an invertible matrix P such that P⁻¹ A P is diagonal; and for a real symmetric matrix A, find an orthogonal matrix Q such that Q^T A Q is diagonal.
  • Construct a mathematical argument in order to deduce or prove simple facts about vectors, matrices, vectors spaces and linear maps.

  
  

  ATTRIBUTES

  At the end of this module, students should have developed with respect to the following attributes:

  • Grasp the principles and practices of their field of study.
  • Acquire substantial bodies of new knowledge.
  • Explain and argue clearly and concisely.
  • Acquire and apply knowledge in a rigorous way.
  • Connect information and ideas within their field of study.
  • Adapt their understanding to new and unfamiliar settings.

  
  

• Where to Get Help
  Questions about the content of the module or its organisation should be posted in the student forum, which will be regularly monitored by the module leaders. You are also encouraged to attend any of the module leaders' learning support hour. For questions regarding personal matters, you can send an email directly to any of the module leaders.
  

• Q-Review
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    Q-Review Recordings External tool