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
    No significant annotations were made
    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 2 as a Hilbert space
    have seen the link between orthonormal bases and isomorphisms from 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 learnt about linear functionals
    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
    Adjoints
    Aims
    have seen different types of operator on Hilbert spaces
    have seen the finite dimensional spectral theorem
    Lecture Notes
    beamer
    trans
    handout
    No significant annotations made
    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
    No significant annotations made
    Resources
    Hilbert Spaces
    External Resources

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