
MAST10007
Linear Algebra
Stream 2

Semester II 2023 
Lecturer for Stream 2: Arun Ram, 174 Richard Berry, phone: 8344 6953, email: aram@unimelb.edu.au
Time and Location:
 Stream 1: M1011 JH Michell,W1011 JH Michell, F1011 JH Michell
 Stream 2: M121 Glyn Davis B117,Th121 Glyn Davis B117, F121 Carrillo Gantner Sidney Meyer
 Stream 3: M14:1515:15 JH MIchell, W14:1515:15 JH MIchell,F14:1515:15 JH Michell
Arun Ram's consultation hours are Tuesdays 9:0010:00, Wednesdays 11:0012:00,
and Thursdays 11:0012:00 in Room 174 of Richard Berry.
The student representatives are
????? email: ???@student.unimelb.edu.au and
???? email: ????@student.unimelb.edu.au
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

𝔽modules with bilinear form: Gram matrices and dual bases

𝔽modules with bilinear form: Orthogonal projections

𝔽modules with bilinear form: Orthonormal sequences and GramSchmidt
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/2011/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
Lectures this semester
 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