Topic outline

  • General

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    • Linear Algebra (in addition to Calculus/Analysis) is the most important part of any university Mathematics course. This 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 pure and applied mathematics, probability and statistics modules. 

      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 eg calculating powers of matrices, 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 eg Least squares estimates.
    • Announcements Forum
    • Student forum

      Forum Description: This forum is available for everyone to post messages to. Students can raise questions or discuss issues related to the module. Students are encouraged to post to this forum and it will be checked daily by the module leaders. Students should feel free to reply to other students if they are able to.

    • Lecture notes File

      These are the typeset lecture notes for the whole module and are essentially our textbook. I may modify them slightly as we go along, in order to correct typos or improve the exposition of the material.

      Material covered each week will be indicated in weekly tabs below, where you will also find handwritten weekly lecture notes.

      Although the lecture notes will define the module's assessable content, there are many, many books and online resources on linear algebra available. Two recommended books (of which there are numerous versions) are:

      • S. J. Leon, Linear Algebra with Applications.

      • H. Anton and C. Rorres, Elementary Linear Algebra: Applications Version.

      You do not need to buy any books. There are more than enough resources in the library and/or online.


      587.7 KB
    • Optional: pre-recorded lectures from 2022/23 Page

  • Module description

  • Syllabus

    1. Revision of systems of linear equations and 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 eg calculating powers of matrices, 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 eg Least squares estimates.

  • Week 1 - Review of Vectors and Matrices

  • Week 2 - Vector spaces

    • WEEK 2 - LECTURES:

      We define vector spaces and discuss their basic properties. We give a large number of examples, including the space \( \mathbb{R}^{m\times n} \) of matrices, the space \(\mathbb{R}^n\) of column vectors, the space \(P_n\) of polynomials and the space \( C[a,b]\) of continuous real-valued functions. We define subspaces and show a number of examples of subspaces.

      WEEK 2 TASKS

      1. Follow the lectures online or in person. Some topics may be done in more detail in pre-recorded lectures.
      2. Read Sections 4.1 and 4.2 in the typeset Lecture notes.
      3. Attend the online tutorial.
      4. Do Coursework 1 on WebWork. 


    • Handwritten notes Week 2 File
      2.3 MB
    • Coursework 1 File
      89.4 KB
    • Coursework 1 solutions File
      117.4 KB

  • week 3 - Span

    • WEEK 3 - LECTURES:

      Span of a set of vectors, definition and basic properties, examples. Spanning set terminology. Span membership test, Spanning set test, with worked examples. Intersection and sum of subspaces.

      WEEK 3 TASKS

      1. Follow the lectures online or in person. Some topics may be done in more detail in pre-recorded lectures.
      2. Read Section 4.3 in the typeset Lecture notes.
      3. Attend the online tutorial.
      4. Do Coursework 2 on WebWork. 

    • Handwritten notes Week 3 File
      2.2 MB
    • Coursework 2 File
      108.2 KB
    • Coursework 2 solutions File
      115.9 KB

  • week 4 - Linear Independence, basis, dimension

    • WEEK 4 - LECTURES:

      Linear independence, definition and basic properties, examples. Linear Independence Test, with worked examples. Basis, definition, properties, examples. Steinitz' Theorem, dimension of a vector space. 

      WEEK 4 TASKS

      1. Follow the lectures online or in person. Some topics may be done in more detail in pre-recorded lectures.
      2. Read Sections 4.4, 4.5, 4.6 in the typeset Lecture notes.
      3. Attend the online tutorial.
      4. Do Coursework 3 on WebWork. 

    • Handwritten notes Week 4 File
      2.9 MB
    • Coursework 3 File
      116.1 KB
    • Coursework 3 solutions File
      122.7 KB

  • Week 5 - Coordinates, Linear Transformations

    • WEEK 5 - LECTURES:

      Coordinates with respect to a basis, transition matrices. Linear transformations, examples and basic properties. 

      WEEK 5 TASKS

      1. Follow the lectures online or in person. Some topics may be done in more detail in pre-recorded lectures.
      2. Read Sections 4.7, 5.1, 5.2 in the typeset Lecture notes.
      3. Attend the online tutorial.
      4. Do Coursework 4 on WebWork. 

    • Handwritten notes week 5 File
      2.8 MB
    • Coursework 4 File
      105.1 KB
    • Coursework 4 solutions File
      111.6 KB

  • week 6 - Linear transformations

    • WEEK 6 - LECTUREs:

      Linear transformations. Kernel and Image. Rank and nullity for linear maps. Rank and nullity for matrices. Associated matrix. 

      WEEK 6 TASKS


      1. Follow the lectures online or in person. Some topics may be done in more detail in pre-recorded lectures.
      2. Read Sections 5.1--5.4, 5.6, 5.8, 4.8, 5.9 in the typeset Lecture notes.
      3. Attend the online tutorial.
      4. Do Coursework 5 on WebWork.

    • Handwritten notes Week 6 File
      5.8 MB
    • Coursework 5 File
      90.8 KB
    • Coursework 5 solutions File
      108.7 KB

  • Week 7 - Reading week

    • Reading week - no lectures or tutorials

  • Week 8, Change of Basis, Isomorphisms

    • WEEK 8 - LECTURES:

      Change of basis formula. Isomorphisms between vector spaces.

      WEEK 8 TASKS:

      • Follow the lectures online or in person. Some topics may be done in more detail in pre-recorded lectures.
      • Read Sections 5.7, 5.10, (5.11 postponed) in the typeset Lecture notes.
      • Attend the online tutorial.
      • Do Coursework 6 on WebWork.

    • Handwritten notes Week 8 File
      4.2 MB
    • Coursework 6 File
      94.0 KB
    • Coursework 6 solutions File
      132.7 KB

  • week 9 - Eigenvalues and eigenvectors

    • WEEK 9 - LECTURES:

      Eigenvalues and eigenvectors. Characteristic polynomial of a matrix and of a linear map. Diagonalisation. 

      WEEK 9 TASKS:

      • Follow the lectures online or in person. Some topics may be done in more detail in pre-recorded lectures.
      • Read Sections 6.1, 6.2, and 7.1 in the typeset Lecture notes.
      • Attend the online tutorial.
      • Do Coursework 7 on WebWork.

    • Handwritten notes Week 9 File
      7.1 MB
    • Coursework 7 File
      109.8 KB
    • Coursework 7 solutions File
      284.0 KB

  • week 10 - Orthogonality

    • WEEK 10 - LECTUREs: 

      Orthogonality, complements, orthonormal sets. Orthogonal matrices. Gram-Schmidt process. 

       

      WEEK 10 TASKS:


      • Follow the lectures online or in person. Some topics may be done in more detail in pre-recorded lectures.
      • Read Sections 7.1--7.4 and 7.6 in the typeset Lecture notes.
      • Attend the online tutorial.
      • Do Coursework 8 on WebWork.

    • Handwritten notes week 10 File
      5.5 MB
    • Coursework 8 File
      97.4 KB
    • Coursework 8 solutions File
      101.7 KB

  • Week 11-Orthogonal projections, spectral theorem, least squares

    • WEEK 11 - LECTUREs: 

      Orthogonal projection, spectral theorem for symmetric matrices, best approximation, least squares problems.

       

      WEEK 11 TASKS:


      • Follow the lectures online or in person. Some topics may be done in more detail in pre-recorded lectures.
      • Read Sections 7.8, 7.5, 7.7 and 5.11 (postponed from Week 8) in the typeset Lecture notes.
      • Attend the online tutorial.
      • Do Coursework 9 on WebWork.

    • Handwritten notes week 11 File
      6.1 MB
    • Coursework 9 File
      131.5 KB
    • Coursework 9 solutions File
      121.6 KB

  • Week 12 - Revision and Applications of Linear Algebra

  • Assessment information

    • Assessment Pattern -  20% Coursework through Webwork, 80% in-person final exam. 

      Format and dates for the in-term assessments - Coursework on Webwork, 10 courseworks submitted on 5 deadlines: 

      1. Set 1+Set 2, submitted on 18 October
      2. Set 3+Set 4, submitted on 01 November
      3. Set 5+Set 6, submitted on 22 November
      4. Set 7+Set 8, submitted on 06 December
      5. Set 9+Set 10, submitted on 20 December

      Format of final assessment - in-person, under exam conditions.

      link to past papers - See past papers under "Assessment" tab below, and note that in 2023/24 we are going back to the format of 2020 and earlier exam papers. Alternatively, use the search facility: https://qmplus.qmul.ac.uk/mod/data/view.php?id=2443216

      Description of Feedback - Webwork sets release solutions immediately after the submission deadline. Non-Webwork Courseworks solutions are uploaded immediately after the submission deadline. Solutions are discussed in Seminar/tutorial/lectures. Lecturer can provide personalised feedback in office hours. 

       

  • Assessment

    Assessment for MTH5112 consists of:


    • ten WebWork online courseworks, submitted on FIVE submission deadlines, worth 20% in total;
    • a final exam worth 80%. 

    • EXAMINABLE MATERIAL: Everything covered in handwritten lecture notes and pre-recorded lectures.

    • PREPARATION FOR THE EXAM: you should carefully work through the lecture notes, coursework and WeBWork exercises. 

      In my opinion, the best route to success is to prepare by carefully studying the handwritten lecture notes, comparing them to the typeset notes, going through coursework material with the goal of achieving a deep level of understanding of the material. This may require several passes through the material, and you have to test yourself periodically with notes closed to ensure that you can state definitions and theorems correctly. Following that, I would recommend practice using past exam papers, and several of them under exam conditions.


    • Solutions to past papers: 

      It is School policy that solutions to past exams generally NOT be provided. 

      To be updated: I solved "Alternative Assessment 2021" in the Revision Lectures (see Recording 11 of the Monday lecture for MTH5212 and Recording 11 of the Thursday lecture for MTH5112). 

      The solutions of "Alternative Assessment 2020" are given below. 

      Other solutions  will NOT be made readily available since I think students focus too much on those and believe that reading past paper solutions constitutes a preparation for the exam. Past exam solutions are not a secret, but I believe it is much more beneficial for students to work on the solutions themselves, and then I will be happy to discuss any serious attempts of students’ solutions. I would invite students to use the scheduled office hours to discuss their solutions.

    • January Exam 2024 File
      126.0 KB
    • Alternative Assessment 2021 File
      102.1 KB
    • Alternative Assessment 2020 File
      94.2 KB
    • Solutions to Alternative Assessment 2020 File
      12.5 MB
    • Mock Alternative Assessment 2020 File
      101.3 KB
    • 2020 Exam File
      106.4 KB
    • 2019 Exam File
      106.6 KB
    • 2018 Exam File
      91.5 KB
    • 2017 Exam File
      86.0 KB
    • 2016 Exam File
      142.0 KB
    • 2015 Exam File
      83.5 KB
    • 2014 Exam File
      146.0 KB

  • 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 QT 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.


  • Q-Review

  • Generated by Assessment Information block

    • Assessment for MTH5112 consists of:


      • TEN WebWork online courseworks, submitted on FIVE submission deadlines, worth 20% in total:

        1. Set 1+Set 2, submitted on 18 October
        2. Set 3+Set 4, submitted on 01 November
        3. Set 5+Set 6, submitted on 22 November
        4. Set 7+Set 8, submitted on 06 December
        5. Set 9+Set 10, submitted on 20 December


      • a final exam worth 80%, in-person, under exam conditions. 


  • General course materials

    • Lecture notes File

      These are the typeset lecture notes for the whole module and are essentially our textbook. I may modify them slightly as we go along, in order to correct typos or improve the exposition of the material.

      Material covered each week will be indicated in weekly tabs below, where you will also find handwritten weekly lecture notes, pre-recorded lectures and synchronous lectures and their recording.

      Although the lecture notes will define the module's assessable content, there are many, many books and online resources on linear algebra available. Two recommended books (of which there are numerous versions) are:

      • S. J. Leon, Linear Algebra with Applications.

      • H. Anton and C. Rorres, Elementary Linear Algebra: Applications Version.

      You do not need to buy any books. There are more than enough resources in the library and/or online.


      587.7 KB

  • Exam papers: availabable under "Assessment information"

  • Reading List Online