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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:

  1. Stream 1: M10-11 JH Michell,W10-11 JH Michell, F10-11 JH Michell
  2. Stream 2: M12-1 Glyn Davis B117,Th12-1 Glyn Davis B117, F12-1 Carrillo Gantner Sidney Meyer
  3. Stream 3: M14:15-15:15 JH MIchell, W14:15-15:15 JH MIchell,F14:15-15:15 JH Michell

Arun Ram's consultation hours are Tuesdays 9:00-10:00, Wednesdays 11:00-12:00, and Thursdays 11:00-12:00 in Room 174 of Richard Berry.

The student representatives are
?????   email: ???@student.unimelb.edu.au   and
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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
  • Normal form summary

Solving systems of linear equations

  • Kernels and Images
  • Solutions of Systems of linear equations
  • 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 projections
  • Cauchy-Schwarz, the Triangle inequality and angles between vectors
  • Hamiltonians, or space-time, and cross products
  • Areas and Volumes of parallelopipeds
  • Equations of lines and planes in ℝ3

Vector spaces and Linear transformations

  • 𝔽-modules: Vector spaces and linear transformations
  • 𝔽-modules: Some proofs
  • 𝔽-modules: Bilinear forms, sesquilinear forms and quadratic forms

𝔽-modules with bilinear form

  • 𝔽-modules with bilinear form: Gram matrices and dual bases
  • 𝔽-modules with bilinear form: Orthogonal projections
  • 𝔽-modules with bilinear form: Orthonormal sequences and Gram-Schmidt

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 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

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: 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