
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 14:1515:15 JH Michell, Wednesday 14:1515:15 JH Michell, Friday 14:1515:15 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 12:151:45 in Evan Williams Theatre, Peter Hall Building.
Attempted recording of these session is as follows:
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
Normal forms
Solving systems of linear equations
Matrix groups
Flag varieties
Determinants
Eigenvalues, eigenvectors and diagonalization
Vector Geometry
Vector spaces and Linear transformations
𝔽modules with bilinear form
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: ℝ^{n}, 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, 3dspacetime
Hand written Lecture Notes
 11 and 14 August 2023 Lecture:
Lines and planes in ℝ^{3}
Hand written Lecture Notes
 16 August 2023 Lecture:
Fields, 𝔽vector spaces and 𝔽subspaces
Hand written Lecture Notes
 18 August 2023 Lecture:
𝔽subspaces, 𝔽span, 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:ℝ^{2}>ℝ^{2}
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
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 nspace, 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: CayleyHamilton 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: 𝔽modules and 𝔽linear 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: CauchySchwarz and triangle inequalities
 Week 9: Projections and GramSchmidt
 18 September 2016 Lecture 25: Gram matrices and dual bases
 20 September 2016 Lecture 26: Projections
 22 September 2016 Lecture 27: Orthonormal sequences and GramSchmidt
 Week 10:
ℝ^{2},
ℝ^{3},
and ℝ^{n}
 2 October 2016 Lecture 28:
Lengths, distances and projections in
ℝ^{2},
ℝ^{3},
and ℝ^{n}
 4 October 2016 Lecture 29:
Equations of lines and planes in
ℝ^{2},
ℝ^{3},
and ℝ^{n}
 6 October 2016 Lecture 30: Cross products and spacetime
 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