MathsNotes lecture notes 2011

Lecture Notes

The following is a list of the lectures together with the relevant wiki pages. Additional resources can be found below this list organised by topic.

Also available here are the lecture notes. I shall endeavour to make the notes available the day before the lecture so that students can “follow along”. There may be additional notes written during the lectures. These will be posted shortly after the lecture finishes. I shall often prepare a little more than I shall actually give, any extra will be taken up in the next lecture or deferred to the wiki.

The notes will be available in several different layouts. It is important to know which is which.

Beamer
This is what will actually appear on the screen during the lectures. You must never print this version. As each “transition” results in a new page, this can easily exceed 100 pages.
Trans
This is a “one frame per page” version of the above. If you intend to “follow along” with the lecture on your own computer then this is probably the best. However, I still strongly urge you not to print it.
Handout
This is a more condensed version in terms of space. By putting 4 slides on a page the total number of pages is significantly reduced. If you want to print something then print this version.
Annotations
This PDF contains just the annotated pages from the lecture. For most pages, this should be sufficient to locate it within the main presentation. For some it may be useful to have the previous page included as well. Let me know if this, or something else, would be useful.

1. 24th August 2011: Introduction

Topics
Sets
Aims
have seen an overview of the course
have been told the requirements
have seen the intuitive definition of a set
know how to test if two sets are equal
Lecture Notes
beamer
trans
handout
Resources
set theory
External Resources
Extra
Forum post on the ambiguity of the word “contains”.
2. 26th August 2011: Functions

Topics
Functions
Aims
have seen the intuitive definition of a function
know how to test if two functions are equal
have seen an example of constructing a new function from old ones
Lecture Notes
beamer
trans
handout
No significant annotations were made during this lecture
Resources
set theory
Understanding Proofs
External Resources
3. 31st August 2011: Calculating with Calculators

Topics
Metric Spaces
Aims
have seen how practical considerations lead to the need for the notion of “approximation”
have seen the definition of a metric space
have seen examples of metric spaces
Lecture Notes
beamer
trans
handout
annotations
Resources
Metric Spaces
External Resources
4. 2nd September 2011: Producing Polynomials

Topics
Metric Spaces, Sequences
Aims
have seen how to use an ODE to generate an approximation
have seen how sequences provide lists of approximations
have seen the definition of convergence in a metric space
have seen an example of how to use that definition
Lecture Notes
beamer
trans
handout
annotations
Resources
Metric Spaces
External Resources
5. 7th September 2011: Complete Convergence

Topics
Complete Metric Spaces, Cauchy Sequences
Aims
have seen examples of metric spaces and sequences
have seen how to remove the “limit” from convergence
have seen the definition of a “complete” metric space
have seen examples of complete metric spaces
Lecture Notes
beamer
trans
handout
No annotations were made (technology failure)
Resources
Metric Spaces
External Resources
6. 9th September 2011: Completely fixed

Topics
Banach’s fixed point theorem
Aims
have seen how completeness helps us solve ODEs
have seen how the question of finding approximations leads to Banach’s fixed point theorem
Lecture Notes
beamer
trans
handout
annotations
Resources
Metric Spaces
External Resources
Notes
We did not cover all the material in the documents for this lecture
7. 14th September 2011: From Here to There

Topics
Banach’s fixed point theorem
Continuous maps
Aims
have seen the statement of and examples of Banach’s Fixed Point theorem
have seen the motivation for, and definitions of, a continuous function between metric spaces
Lecture Notes
beamer
trans
handout
annotations
Resources
Metric Spaces
External Resources
8. 16th September 2011: Inheritance

Topics
Continuous maps
Special Subsets
Subsequences
Aims
have seen how useful inheritance is in mathematics
Lecture Notes
beamer
trans
handout
annotations
Resources
Metric Spaces
External Resources
9. 21st September 2011: An Angular Attack

Topics
Inner Products
Norms
Aims
have seen how angles lead to inner products
Lecture Notes
beamer
trans
handout
annotations
Resources
Hilbert Spaces
External Resources
10. 23rd September 2011: That Which We Call a Vector

Topics
Inner Products
Norms
Vector Spaces
Aims
have seen how to make sense of angles between continuous functions
Lecture Notes
beamer
trans
handout
annotations
Resources
Hilbert Spaces
The space of continuous functions
External Resources
11. 28th September 2011: The Point of Angles

Topics
Inner Products
Norms
Vector Spaces
Aims
have seen how inner products provide us with the tools to make sense of “the best approximation”
Lecture Notes
beamer
trans
handout
annotations
Resources
Hilbert Spaces
External Resources
12. 30th September 2011: The Right Angle

Topics
Inner Products
Norms
Vector Spaces
Aims
have seen an application of the closest point property
have seen the construction of square integrable functions
Lecture Notes
beamer
trans
handout
annotations
Resources
Hilbert Spaces
External Resources
13. 5th October 2011: The Luxury of Linearity

Topics
Linear Transformations
Matrices
Vector Spaces
Aims
have seen the definition of a linear transformation
have seen examples of linear transformations
have seen the basic properties of linear transformations
Lecture Notes
beamer
trans
handout
(For technical reasons, saving the annotations didn’t work for this lecture.)
Resources
Linear Algebra
External Resources
14. 7th October 2011: Dimension Leap

Topics
Linear Transformations
Matrices
Vector Spaces
Aims
have seen the definition of dimension
have seen the basic properties of dimension
have seen how to convert problems about linear transformations to ones about matrices
Lecture Notes
beamer
trans
handout
annotation slide a,annotation slide b
Resources
Linear Algebra
External Resources
15. 12th October 2011: Rank Invariants

Topics
Linear Transformations
Matrices
Vector Spaces
Aims
have seen the notion of “invariant”
have seen how to classify linear transformations
Lecture Notes
beamer
trans
handout
annotations
Resources
Linear Algebra
External Resources
16. 14th October 2011: There and Back Again

Topics
Linear Transformations
Matrices
Eigenvalues/Eigenvectors
Jordan Canonical Form
Aims
have seen how insisting on the same point of view makes life more interesting
why nilpotent linear transformations are useful
how eigenvectors enter the fray
how to classify (most) linear transformations with the same target as source
Lecture Notes
beamer
trans
handout
No significant annotations were made during this lecture.
Resources
Linear Algebra
External Resources
17. 19th October 2011: How to Solve It

Topics
Matrix decomposition
Solving linear systems
Aims
How factorisation of matrices helps solve problems
Why triangular matrices are Very Nice
Lecture Notes
beamer
trans
handout
annotations
Resources
Linear Algebra
External Resources
18. 21st October 2011: The Return of the Angle

Topics
Matrix decomposition
The QR factorisation
Inner products
Aims
See that isomorphisms should respect pre-existing structure
How bringing back inner products leads to isometries …
… and the QR factorisation
Lecture Notes
beamer
trans
handout
annotations
Resources
Linear Algebra
External Resources
Hilbert Spaces
External Resources
19. 26th October 2011: Factors and Angles

Topics
Matrix decomposition
Gram-Schmidt
The QR factorisation
The singular value decomposition
Aims
See some of the nice properties of matrices with orthonormal columns
See a better “orthogonal” factorisation than QR
Lecture Notes
beamer
trans
handout
annotations
Resources
Linear Algebra
External Resources
Hilbert Spaces
External Resources
20. 28th October 2011: Polynomials

Topics
Matrix decomposition
Minimum and characteristic polynomials
The Cayley-Hamilton theorem
Aims
see how ubiquitous polynomials are
learn about the minimum polynomial of a linear transformation
see the true definition of the characteristic polynomial
see the Cayley-Hamilton theorem
Lecture Notes
beamer
trans
handout
annotations
Resources
Linear Algebra
External Resources
Hilbert Spaces
External Resources
21. 2nd November 2011: Everything You Ever Wanted to Know About Hilbert Spaces

Topics
Hilbert spaces
Aims
have seen a glimpse of why Hilbert spaces are so great
have seen some of the most important features of Hilbert spaces
Lecture Notes
beamer
trans
handout
annotations
Resources
Hilbert Spaces
External Resources
22. 4th November 2011: Bases and Isomorphisms

Topics
Hilbert spaces
Aims
have examined ${\ell }^{2}$ as a Hilbert space
have seen the link between orthonormal bases and isomorphisms from ${\ell }^{2}$
Lecture Notes
beamer
trans
handout
annotations
Resources
Hilbert Spaces
External Resources
23. 9th November 2011: Near and Far Away

Topics
Hilbert spaces
Orthogonal Subspaces
Aims
learn about splittings of vector spaces
seen the definition of the orthogonal complement
related the above to closest points
Lecture Notes
beamer
trans
handout
annotations
Resources
Hilbert Spaces
External Resources
24. 11th November 2011: Duality

Topics
Hilbert spaces
Dual spaces
Aims
have seen how duality works with Hilbert spaces
have seen the Riesz representation theorem
Lecture Notes
beamer
trans
handout
annotations
Resources
Hilbert Spaces
External Resources
25. 16th November 2011: Best of both: Continuous and Linear

Topics
Hilbert spaces
Continuous linear transformations
Aims
have examined continuous linear transformations
Lecture Notes
beamer
trans
handout
annotations
Resources
Hilbert Spaces
External Resources
26. 18th November 2011: Spectral Theory

Topics
Hilbert spaces
Continuous linear transformations
Aims
have seen different types of operator on Hilbert spaces
have seen the finite dimensional spectral theorem
Lecture Notes
beamer
trans
handout
Resources
Hilbert Spaces
External Resources
27. 23rd November 2011: The Grand Finale

Topics
Hilbert spaces
Partial differential equations
Aims
have seen an example of Hilbert space theory applied to partial differential equations
Lecture Notes
beamer
trans
handout