Contact | Description | Topics | Texts | Notes | Lectures | Assessment and Assignments |

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MAST10007
Linear Algebra
Stream 3

Semester II 2023

Lecturer for Stream 3: Arun Ram, 174 Peter Hall Building, email: aram@unimelb.edu.au

Time and Location: I am lecturing

  • Stream 3 - Arun Ram: Monday 14:15-15:15 JH Michell, Wednesday 14:15-15:15 JH Michell, Friday 14:15-15:15 JH Michell

I am taking Practice Class P01/03 on Wednesdays at 3:15pm in Peter Hall G09.

Arun Ram's Proof and Solution writing sessions are Tuesdays 12:15-1:45 in Evan Williams Theatre, Peter Hall Building. Attempted recording of these session is as follows:

  • 01 August 2023: https://unimelb.zoom.us/rec/share/71quaebaYn2JjwMnR27J33pq5z4q_ZRIfnDwuRLTd93ymWWz2BlF0y0K7Hg6hppa.Dgmi-MINKspfV-op
    Passcode: 37Gjey@c
  • 08 August 2023: I'm sorry, I forgot to bring my phone that day and the two factor authentification locked me out of Zoom so I was not able to make a recording on 0 August.
  • 15 August 2023: https://unimelb.zoom.us/rec/share/RI8Yu5OPhpwRrr8Dwr7FAFZhsA6-x7c10bLZ35kmX0nOjfwKptebKHQeX--oAsAR.G8eQ27Ekhse9nOyN
    Passcode: ^7g4B.$Q
  • 22 August 2023: https://unimelb.zoom.us/rec/share/VX2H0aT1tfXFkDhBo6lF5iPWBm7ibu1D4Hf4vqTEVuQNMeySMBA21Nzg2AYR25uW.HbFkPoA6hEP3KJId
    Passcode: 1k8&=Yk8
  • 29 August 2023: https://unimelb.zoom.us/rec/share/XEUCF5X5NpUAHD320sZxZb6tZcIr8VwlVkK9jZ1RNEBqMaM9V6VY_ZMqic2aRUoT.h5GvcWeEipq84Nib
    Passcode: B@eX!y3v
  • 5 September 2023: https://unimelb.zoom.us/rec/share/u_fWEMASBwqLNo8M_Xm-8BxfJYj8NKeh5DOGlrnr9hMu-hNvaOxdCGIzmlJwt4ic.C_B5G_MUtv81LzgV
    Passcode: $3bJc8*W
  • 12 September 2023: The computer in the room would not access the room camera, so I was unable to make a recording.
  • 19 September 2023: https://unimelb.zoom.us/rec/share/mxWMxUyNHVWFKKp_wad96NSUb4PZfv4zYvWznNSbGBiV8xBhkM9f_CiVglH5XjWc.o0AiKqEZHjIrhGfi
    Passcode: YZiHs&3O
  • 3 October 2023: https://unimelb.zoom.us/rec/share/ZddZHUP9qatlMW0b6IaH-9qehlLjzQddgEISLGNDAi6HHw9xzLjVgNSPshDoqveA.HlIjEqMCSJBHO-GQ
    Passcode: ?3z6N*G1
  • 10 October 2023: https://unimelb.zoom.us/rec/share/hCXaNSfxAKDGMcgHZ5WUYWZ_DqYUiQz9pBlfn3eFaLXwAUM3wejySTQLcyjFSNAI.VTMwgXOJsx3djnHy
    Passcode: ECui*eW8
  • 17 October 2023: https://unimelb.zoom.us/rec/share/Kt91e_12fHyAJr8NIIjOG2hkTRpNVb4_UdU5XyyvqbLV6qkGIZ83Vxci-rLXltUo.7IjhdEaVK0HxoOCn
    Passcode: i=Q56y1G

I am often available for additional questions/discussions after class on Monday and after the Practical that I take on Wednesday. Follow us to my office to continue the discussion after Monday class or Wednesday Prac. Most Fridays I attend the Pure Mathematics seminar at 3:15 in Room 162 of Peter Hall.

I am available by appointment. I rarely come in to the University before 12 as I have other responsibilities in the morning until 12. If you email me suggesting some weekday afternoon times for an appointment that work for you then I can choose one of them that also works for me. If you email me and don't suggest some times that work for you, then I will respond by asking you to suggest some weekday afternoon times that work for you.

Some of my thoughts about teaching have been written down in the Lecture script

Teaching Math in the Next Life

Here is a guide to Proof Machine with examples.

Proof Machine

Notes written by Arun Ram

Matrices and operations

  • Matrices and operations
  • Matrices and operations: Some proofs
  • Matrices and operations: Some examples

Normal forms

  • Generators
  • Examples of the steps in the normal form algorithm
  • The normal form algorithm
    • Topic 2: Examples 1 and 2 and 3 and 4
    • Topic 2: Example 5
    • Topic 2: Example 7 and 8 and 9
    • Topic 2: Examples 10 to 13
    • Topic 2: Examples 14 to 16
  • Normal form summary

Solving systems of linear equations

  • Kernels and Images
  • Solutions of Systems of linear equations
    • Topic 1: Examples 2 and 3 and 4
    • Topic 1: Example 6
    • Topic 1: Example 7 and 8
    • Topic 1: Example 9
    • Topic 1: Example 10
    • Topic 1: Example 11
  • Linear Systems: Some Proofs

Matrix groups

  • Diagonal matrices
  • Permutation matrices
  • Unipotent upper triangular matrices
  • Invertible matrices
  • Matrix group presentations: Some proofs

Flag varieties

  • Bruhat decomposition

Determinants

  • Determinants are homomorphisms
  • Determinants: The permutation formula and Laplace expansion
  • Uses of determinants: Inverses, Cramer's rule and the Cayley-Hamilton
  • Determinants: Some proofs

Eigenvalues, eigenvectors and diagonalization

  • Eigenvalues and eigenvectors
  • Ordered bases and ordered orthonormal bases
  • Diagonalization
  • Some proofs

Vector Geometry

  • ℝ2 and ℝn, lengths, distances and the standard inner product
  • Angles, orthogonality and projections
  • 3d-space-time and cross products
  • Determinants and volumes
  • Equations of lines and planes in ℝ3
    • Topic 3 Examples 1 and 2 and 4
    • Topic 3 Example 5
    • Topic 3 Example 6 and 7
    • Topic 3 Examples 8 and 9
    • Topic 3 Examples 10 and 11 and 12
    • Topic 3 Example 13

Vector spaces and Linear transformations

  • 𝔽-modules: Vector spaces and linear transformations
  • 𝔽-modules: Some proofs
    • Subspaces: Topic 4 Examples 7 and 8
    • Subspaces: Topic 4 Examples 9 and 11
    • Subspaces: Topic 4 Example 10
    • span: Topic 4 Examples 13 and 14 and 15
    • span: Topic 4 Example 16
    • span: Topic 4 Example 17
    • span: Topic 4 Example 18
    • Linear independence: Topic 4 Example 19
    • Linear independence: Topic 4 Examples 20 and 22
    • Linear independence: Topic 4 Example 21
    • Linear independence: Topic 4 Example 23
  • 𝔽-modules: Bilinear forms, sesquilinear forms and quadratic forms

𝔽-modules with bilinear form

  • Bilinear forms
  • Gram matrices and Cauchy-Schwarz
  • Nondegeneracy and dual bases
  • Orthogonal decompositions
  • The Gram-Schmidt process
  • Adjoint of a linear transformation and matrix
  • The Spectral theorem
  • Some proofs: Bilinear forms

Announcements

  • No books, notes, calculators, ipods, ipads, phones, etc at the exam.
  • Prof. Ram reads email but generally does not respond by email. Usually these are collated and reponses to email queries are provided in the first few minutes of lectures. That way all students can benefit from the answer to the query.

In class lectures

  • 28 July 2023 Lecture: Matrix operations and Elementary Matrices. Hand written Lecture Notes

  • 31 July 2023 Lecture: Solving linear systems and writing matrices and inverse as products of elementary matrices Hand written Lecture Notes

  • 2 August 2023 Lecture: Properties of inverses and writing noninvertible matrices as products of elementary matrices Hand written Lecture Notes

  • 4 August 2023 Lecture: Rank, determinants and transpose Hand written Lecture Notes

  • 7 August 2023 Lecture: ℝn, lengths, standard inner product, angles, projections, and the cofactor formula for the dterminant Hand written Lecture Notes

  • 9 and 11 August 2023 Lecture: Perpendicular and parallel, Volumes from determinants, cross products, 3d-space-time Hand written Lecture Notes

  • 11 and 14 August 2023 Lecture: Lines and planes in ℝ3 Hand written Lecture Notes

  • 16 August 2023 Lecture: Fields, 𝔽-vector spaces and 𝔽-subspaces Hand written Lecture Notes

  • 18 August 2023 Lecture: 𝔽-subspaces, 𝔽-span, Linear combinations, linear independent, and Problem solving by rewriting the question. Hand written Lecture Notes

  • 21 August 2023 Lecture: Linear independence, Span, Proof Machine Hand written Lecture Notes

  • 23 August 2023 Lecture: Span, Equal sets Hand written Lecture Notes

  • 25 August 2023 Lecture: The Vector space axioms Hand written Lecture Notes

  • 28 August 2023 Lecture: Bases and the MinMax theorem Hand written Lecture Notes

  • 30 August 2023 Lecture: Solution space (i.e. kernel), Column space (i.e. image) and Row Space of a matrix Hand written Lecture Notes

  • 1 September 2023 Lecture: Favourie bases of favourite vectors spaces and more examples of Solution space (i.e. kernel), Column space (i.e. image) and Row Space of a matrix Hand written Lecture Notes

  • 4 September 2023 Lecture: Linear transformations, kernel, image and matrices with respect to a basis for the source and a basis fot he target Hand written Lecture Notes

  • 6 September 2023 Lecture: Pictures of Geometric linear transformations T:ℝ2->ℝ2 Hand written Lecture Notes

  • 8 September 2023 Lecture: Kernel, image, injective, surjective Hand written Lecture Notes

  • 11 September 2023 Lecture: Eigenvalues, Eigenvectors and Diagonalization Hand written Lecture Notes

  • 13 September 2023 Lecture: More Eigenvalues, Eigenvectors and Diagonalization Hand written Lecture Notes

  • 15 September 2023 Lecture: Review of change of basis matrices, and kernel and image of a linear transformation Hand written Lecture Notes

  • 18 September 2023 Lecture: A genetics Markov chain, and eigenvectors of the derivative Hand written Lecture Notes

  • 21 September 2023 Lecture: Symmetric, Hermitian, Orthogonal and Unitary diagonalisation Hand written Lecture Notes

  • 2 October 2023 Lecture: Inner products, lengths, distances, angles Hand written Lecture Notes

  • 4 October 2023 Lecture: Gram matrices and checking the axioms Hand written Lecture Notes

  • 6 October 2023 Lecture: Orthogonal sets and orthonormal sets Hand written Lecture Notes

  • 9 October 2023 Lecture: Projections onto subsapces Hand written Lecture Notes

  • 11 October 2023 Lecture: Line of best fit and Gram-Schmidt Hand written Lecture Notes

  • 13 October 2023 Lecture: Singular value decomposition Hand written Lecture Notes

  • 16 October 2023 Lecture: Hermitian, Unitary, symmetric and orthogonal matrices Hand written Lecture Notes

  • 18 October 2023 Lecture: Revision topics Hand written Lecture Notes

  • 20 October 2023 Lecture: Revision topics Hand written Lecture Notes

Subject Outline

The handbook entry for this course is at https://handbook.unimelb.edu.au/view/2023/MAST10007. The subject overview that one finds there:

This subject gives a solid grounding in key areas of modern mathematics needed in science and technology. It develops the concepts of vectors, matrices and the methods of linear algebra. Students should develop the ability to use the methods of linear algebra and gain an appreciation of mathematical proof. Little of the material here has been seen at school and the level of understanding required represents an advance on previous studies.

Systems of linear equations, matrices and determinants; vectors in real n-space, cross product, scalar triple product, lines and planes; vector spaces, linear independence, basis, dimension; linear transformations, eigenvalues, eigenvectors; inner products, least squares estimation, symmetric and orthogonal matrices.

Main Topics

  • (1) Matrices
  • (2) Normal forms for matrices
  • (3) Solving linear equations
  • (4) Determinants
  • (5) Eigenvalues and eigenvectors
  • (6) Vector spaces and linear transformations
  • (7) Bilinear forms and quadratic forms

Assessment

Follow the LMS for this.

Resources

  • The exercise book for the course
  • The tutorial sheets for the course
  • The exercise book for the course
  • The notes on this page
  • The exercise book for the course
  • Any of hundreds of books on linear algebra accessible from the internet and the University of Melbourne library
  • The exercise book for the course

A Dream Lecture by Lecture Plan

  • Week 1: Matrices
    • 24 July 2016 Lecture 1: Matrices
    • 26 July 2016 Lecture 2: Theorems about matrix operations
    • 28 July 2016 Lecture 3: Normal forms for matrices: generators

  • Week 2: Normal forms
    • 31 August 2016 Lecture 4: Normal forms for matrices: the algorithm
    • 2 August 2016 Lecture 5: Normal forms for matrices: theorems and examples
    • 4 August 2016 Lecture 6: Invertible matrices and normal form

  • Week 3: Solving systems of linear equations
    • 7 August 2016 Lecture 7: Kernels and images
    • 9 August 2016 Lecture 8: Solving systems of linear equations
    • 11 August 2016 Lecture 9: Solving systems of linear equations

  • Week 4: Matrix groups
    • 14 August 2016 Lecture 10: Diagonal and permutation matrices
    • 16 August 2016 Lecture 11: Unipotent upper triangluar and invertible matrices
    • 18 August 2016 Lecture 12: Determinants: Classification of multiplicative maps

  • Week 5: Determinants
    • 21 August 2016 Lecture 13: The permutation formula for determinants
    • 23 August 2016 Lecture 14: Laplace expansion, inverse by determinants, Cramer's rule
    • 25 August 2016 Lecture 15: Cayley-Hamilton theorem and traces

  • Week 6: Eigenvalues and diagonalization
    • 28 August 2016 Lecture 16: Eigenvalues and eigenvectors
    • 30 August 2016 Lecture 17: Diagonalization (and spectral theorem)
    • 1 September 2016 Lecture 18: (Diagonalization and) spectral theorem

  • Week 7: Vector spaces and linear transformations
    • 4 September 2016 Lecture 19: 𝔽-modules and 𝔽-linear maps
    • 6 September 2016 Lecture 20: Kernels and images
    • 8 September 2016 Lecture 21: Bases and dimension

  • Week 8: Bilinear forms and quadratic forms
    • 11 September 2016 Lecture 22: Matrices of linear transformations and change basis
    • 13 September 2016 Lecture 23: Bilinear forms and quadratic forms
    • 15 September 2016 Lecture 24: Cauchy-Schwarz and triangle inequalities

  • Week 9: Projections and Gram-Schmidt
    • 18 September 2016 Lecture 25: Gram matrices and dual bases
    • 20 September 2016 Lecture 26: Projections
    • 22 September 2016 Lecture 27: Orthonormal sequences and Gram-Schmidt

  • Week 10: ℝ2, ℝ3, and ℝn
    • 2 October 2016 Lecture 28: Lengths, distances and projections in ℝ2, ℝ3, and ℝn
    • 4 October 2016 Lecture 29: Equations of lines and planes in ℝ2, ℝ3, and ℝn
    • 6 October 2016 Lecture 30: Cross products and space-time

  • Week 11: Review and revision
    • 9 October 2016 Lecture 31: Examples
    • 11 October 2016 Lecture 32: More examples
    • 13 October 2016 Lecture 33: and More examples

  • Week 12: Review and revision
    • 16 October 2016 Lecture 34: Sample exam questions
    • 18 October 2016 Lecture 35: More sample exam questions
    • 20 October 2016 Lecture 36: and More sample exam questions