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MAST10007
Linear Algebra
Summer Term

Summer 2026

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

Time and Location: I am coteaching with Gufang Zhao

  • Tuesday 9:00-10:00, 12:00-13:00, 15:00-16:00 and Thursday 9:00-10:00, 12:00-13:00, 15:00-16:00 and PAR-133-B1-B117-B117 Theatre & Event Venue (510)

Arun Ram's consultation hours are Tuesdays and Thursdays 10:00-12:00 in Room 174 of the Peter Hall Building starting on 8 January.

I am often available for additional questions/discussions after class.

I am available by appointment. If you email me suggesting some 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

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.

Subject Outline

The handbook entry for this course is at https://handbook.unimelb.edu.au/view/2026/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

In class lectures from 2026

  • 9am on 6 January 2026 Lecture: Matrix operations
    Handwritten Lecture Notes, Additional lecture Slides, Matrices and operations notes, Matrices and operations proofs, Matrices and operations examples,

    Student TODO List:
    • Carefully read all the questions in Topic 2 of the exercise booklet.
    • Do questions 15 16, 17, 18, 19, 21, 22 of Topic 2 in the exercise booklet.
    • Carefully work through all the Matrix operation examples in the Matrix Operations examples notes.
    • Then read through the Matrices and operations notes, and figure out where each of the definitions and results is on the main lecture slides for the course.
    • Then go through the Matrices and operations proofs notes so that you know exactly where to find the proof of each result about matrix operations that appears on the main lecture slides.
    • Then go through the Additional lecture Slides and make a note of how these might be improved and pass on your suggestions to Prof. Ram (by email, or before or after class).
    • Do the Pop quiz on lecture 1 material enough times that if it actually gets given as a pop quiz then you get a high mark on it.

  • 12noon on 6 January 2026 Lecture: Greedy normal form
    Handwritten Lecture Notes, Additional lecture Slides, Inverses and normal forms Notes, Normal form with example and proof Notes,

    Student TODO List:
    • Carefully read all the questions in Topic 2 of the exercise booklet.
      For any words or terms in these questions that you do not understand, ask your tutor (or in class) what they mean.
    • Do questions 23, 24, 25, 26, 28 of Topic 2 in the exercise booklet.
    • Read Secion 2.2.1 of Inverses and normal forms Notes,
    • Read section 2.4.1 of Normal form with example and proof Notes,
    • Check Section 2.4.3 of Normal form with example and proof Notes, line by line
    • Do the Pop quiz on lecture 2 material enough times that if it actually gets given as a pop quiz then you get a high mark on it.

  • 3pm on 6 January 2026 Lecture: Solutions of linear systems
    Handwritten Lecture Notes, Additional lecture Slides, Kernels, Images and Sol(Ax=b) Notes,

    Student TODO List:
    • Carefully read all the questions in Topic 2 of the exercise booklet.
      For any words or terms in these questions that you do not understand, ask your tutor (or in class) what they mean.
    • Carefully read all the questions in Topic 1 of the exercise booklet.
      For any words or terms in these questions that you do not understand, ask your tutor (or in class) what they mean.
    • Do questions 29, 30, 31, 32 of Topic 2 in the exercise booklet.
    • Do questions 3,4,5,6 of Topic 1 in the exercise booklet.
    • Read Section 2.3.2 of Kernels, Images and Sol(Ax=b) Notes,
    • Check Section 2.4.4 and 2.4.5 of Normal form with example and proof Notes, line by line
    • Do the Pop quiz on lecture 3 material enough times that if it actually gets given as a pop quiz then you get a high mark on it.

  • 9am on 8 January 2026 Lecture: Solutions of linear systems
    Handwritten Lecture Notes, Additional lecture Slides,

    Student TODO List:
    • We got behind the original plans for Tuesday, so the TODO list for this lecture has the same key points as for the 3pm 6 January lecture.
      For any words or terms in these questions that you do not understand, ask your tutor (or in class) what they mean.
    • Carefully read all the questions in Topic 1 of the exercise booklet.
      For any words or terms in these questions that you do not understand, ask your tutor (or in class) what they mean.
    • Do questions 29, 30, 31, 32 of Topic 2 in the exercise booklet.
    • Do questions 3,4,5,6 of Topic 1 in the exercise booklet.
    • Do the Pop quiz on lecture 4 material enough times that if it actually gets given as a pop quiz then you get a high mark on it.

  • 12noon on 8 January 2026 Lecture: Row operations
    Handwritten Lecture Notes, Additional lecture Slides,

    Student TODO List:
    • Do questions 7,8,9,10,11,12,13,14 of Topic 1 in the exercise booklet.
    • Do the Pop quiz on lecture 5 material enough times that if it actually gets given as a pop quiz then you get a high mark on it.

  • 3pm on 8 January 2026 Lecture: Kernels and Images
    Handwritten Lecture Notes, Additional lecture Slides,

    Student TODO List:
    • Do questions 7,8,9,10,11,12,13,14 of Topic 1 in the exercise booklet.
    • Do questions 29,30,31,32 of Topic 2 in the exercise booklet.
    • Do the Pop quiz on lecture 6 material enough times that if it actually gets given as a pop quiz then you get a high mark on it.

  • 9am on 13 January 2026 Lecture: Determinants
    Handwritten Lecture Notes, Additional lecture Slides,

    Student TODO List:
    • Do questions 34, 35, 36, 37, 38, 39, 40 of Topic 2 in the exercise booklet.
    • Do questions 29,30,31,32 of Topic 2 in the exercise booklet.
    • Do the Pop quiz on lecture 7 material enough times that if it actually gets given as a pop quiz then you get a high mark on it.

  • 12noon on 13 January 2026 Lecture: The Hilbert space ℝn
    Handwritten Lecture Notes, Additional lecture Slides,

    Student TODO List:
    • Do questions 41,42,43,44,45,49,50 of Topic 3 in the exercise booklet.
    • Do the Pop quiz on lecture 8 material enough times that if it actually gets given as a pop quiz then you get a high mark on it.

  • 3pm on 8 January 2026 Lecture: Equations of Lines and planes in 𝕣3
    Handwritten Lecture Notes, Additional lecture Slides,

    Student TODO List:
    • Do questions 55,56,57,58,61 of Topic 3 in the exercise booklet.
    • Do the Pop quiz on lecture 9 material enough times that if it actually gets given as a pop quiz then you get a high mark on it.

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

In class lectures from 2023

  • 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

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