Arun Ram: Linear Algebra Sem II 2016

MAST10007
Linear Algebra
Stream 4

Semester II 2016

Lecturer: Arun Ram, 174 Richard Berry, phone: 8344 6953, email: aram@unimelb.edu.au

Time and Location:
       Lecture Tuesday 11:00-12:00 Redmond Barry Rivett Theatre
       Lecture Thursday 9:00-10:00 Richard Berry JH Michell Theatre
       Lecture Friday 11:00-12:00 Redmond Barry Rivett Theatre

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.

Registration for the SSLC is via online form https://MathsandStatsUoM.formstack.com/forms/sslc_registration_form
The student representatives are
Newaz Saif email: snewaz@student.unimelb.edu.au and
Brandon Hung email: bhung@student.unimelb.edu.au Aditya Mohan email: a.kondamudi@student.unimelb.edu.au

Announcements

No books, notes, calculators, ipods, ipads, phones, etc at the exam.
Prof. Ram reads email but generally does not respond.
The start of semester pack includes: Plagiarism (pdf file), Plagiarism declaration (pdf file), Academic Misconduct (pdf file), SSLC responsibilities (pdf file),
It is University Policy that:
“a further component of assessment, oral, written or practical, may be administered by the examiners in any subject at short notice and before the publication of results. Students must therefore ensure that they are able to be in Melbourne at short notice, at any time before the publication of results” (Source: Student Diary).
Students who make arrangements that make them unavailable for examination or further assessment, as outlined above, are therefore not entitled to an alternative opportunity to present for the assessment concerned (i.e. a ‘make-up’ examination).

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) Linear equations
(2) Matrices and determinants
(3) Euclidean vector spaces
(4) General vector spaces
(5) Linear transformations
(6) Inner products spaces
(7) Eigenvalues and eigenvectors

Assessment

Follow the LMS for this.

Resources part I: recommended Texts

The tutorial sheets for the course
The exercise book for the course
The notes at http://researchers.ms.unimelb.edu.au/~aram@unimelb/notes.html In particular, the following are particularly relevant to this course:
- Lecture 1: Numbers
- Numbers again
- The integers ℤ
- Z/nZ
- The Rationals ℚ
- The quaternions
- Finite fields
- Matrices M_n(F)
- - Numbers - Exercises
  - Z, Q, R, C - Exercises
- Polynomials F[x], F[[x]], F(x) and F((x)); xⁿ
- The rings Q[[x]] and partial fractions
- Sets, functions, cardinality and composition of functions
- Relations
- Ordered sets
- Sets and functions proofs
- - Sets: Examples
  - Functions: Examples
  - Sets: Exercises
  - Functions: Exercises
  - Cardinality: Exercises
  - Ordered sets: Exercises
  - Orders: Exercises
  - Orders on Z,Q, R,C: Exercises
- Fields
- Vector spaces
- Rings
- Modules
- Operations
  - Operations - Exercises
- Groups, rings and fields
  - Groups and monoids - Exercises
- Fields and integral domains (text)
- Fields and integral domains (proofs)
- Fields of fractions
- Fields and ordered fields
- Ordered fields
- Absolute value
  - The triangle and Cauchy Schwartz inequalities
  - Absolute value - Exercises
- Fields with valuation
- Euclidean domains, Principal ideal domains and Unique factorization domains (text)
- Euclidean domains, Principal ideal domains and Unique factorization domains (proofs)
- Polynomial rings (text)
- Polynomial rings (proofs)
Bhattacharya, Jain and Nagpaul, Basic Abstract Algebra, Cambridge University Press, 1994.
M. Artin, Algebra, Prentice-Hall 1991.
Wikipedia

The following problems page may have helpful examples:

http://www.exampleproblems.com/wiki/index.php/Main_Page

Lectures this semester

Week 1: Linear equations and row operations (pages 1-32) (Vocabulary sheet 1)

26 July 2016 Lecture 1: Housekeeping and Proof machine; handwritten lecture notes (pdf file)
Math Grammar: Definitions, Theorems and How to do Proofs (pdf file)
Examples of proofs written in proof machine (pdf file)
28 July 2016 Lecture 2: Linear equations;
29 July 2016 Lecture 3: Row reduction; Numbers of solutions;

Week 2: Matrices (pages 33-59) (Vocabulary sheet 2)

2 August 2016 Lecture 4: Matrices;
4 August 2016 Lecture 5: transposes and inverses;
5 August 2016 Lecture 6: Linear equations by inverse matrices; Rank of a matrix;

Week 3: Determinants, dot products, angles and projections (pages 60-94) (Vocabulary sheet 3)

9 August 2016 Lecture 7: Determinants;
11 August 2016 Lecture 8: Determinants of elementary matrices;
12 August 2016 Lecture 9: Lengths, dot products, cross products;

Week 4: Lines, planes and linear combinations (pages 95-118) (Vocabulary sheet 4)

16 August 2016 Lecture 10: Lines and planes;
18 August 2016 Lecture 11: Linear combinations;
19 August 2016 Lecture 12: Linear independence;

Week 5: Subspaces, span, bases and dimension (pages 119-156) (Vocabulary sheet 5)

23 August 2016 Lecture 13: Subspaces and span;
25 August 2016 Lecture 14: Span and bases;
26 August 2016 Lecture 15: Bases and dimension;

Week 6: Row space, column space, null spaces, image and kernel (pages 157-186) (Vocabulary sheet 6)

30 August 2016 Lecture 16: Row space and column space;
1 September 2016 Lecture 17: column space and image;
2 September 2016 Lecture 18: row space and kernel;

Week 7: Vector spaces and subspaces (pages 187-208) (Vocabulary sheet 7)

6 September 2016 Lecture 19: Vector spaces and subspaces;
8 September 2016 Lecture 20: Span, linear independence, bases and dimension;
9 September 2016 Lecture 21: More examples (polynomials, functions);

Week 8: Linear transformations (pages 209-236) (Vocabulary sheet 8)

13 September 2016 Lecture 22: Linear transformations (2x2 matrices); Linear transformations (mxn matrices);
15 September 2016 Lecture 23: Linear transformations in general; matrix of a linear transformation;
16 September 2016 Lecture 24: image, kernel, rank and nullity;

Week 9: Matrix of linear transformation, inner porducts, lengths, distances and angles (pages 237-261) (Vocabulary sheet 9)

20 September 2016 Lecture 25: Matrix of a linear transformation;
22 September 2016 Lecture 26: inner products;
23 September 2016 Lecture 27: lengths, distances and angles;

Week 10: Orthogonality and Gram-Schmidt (pages 262-286) (Vocabulary sheet 10)

4 October 2016 Lecture 28: Orthogonality and orthonormal bases;
6 October 2016 Lecture 29: Gram-Schmidt;
7 October 2016 Lecture 30: Least-squares line fitting;

Week 11: Eigenvectors and Eigenvalues (pages 287-320) (Vocabulary sheet 11)

11 October 2016 Lecture 31: Definitions;
13 October 2016 Lecture 32: The characteristic polynomial and finding eigenvectors and eigenvalues;
14 October 2016 Lecture 33: Diagonalization and symmetric matrices;

Week 12: Conics and review (pages 321-333) (Vocabulary sheet 12)

18 October 2016 Lecture 34: Conics;
20 October 2016 Lecture 35: Review;
21 October 2016 Lecture 36: Review;