MathsNotes
TMA4115 lecture notes 2012

Lecture Notes

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. 10th January 2012: Complex Numbers

    Topics
    Complex Numbers
    Aims
    have seen an overview of the course
    have been told the requirements
    have seen the definition of a complex number
    have seen the rules for computing with complex numbers
    have seen examples of using those rules
    Lecture Notes
    beamer
    trans
    handout
    annotations
    Resources
    Complex numbers
    pp ix–xiv of the Course book
  2. 11th January 2012: Complex Numbers and Powers

    Topics
    Complex Numbers
    Aims
    have seen how powers work with complex numbers
    have seen a definition of the complex exponential function
    Lecture Notes
    beamer
    trans
    handout
    annotations
    Resources
    Complex numbers
    pp ix–xviii,xxiii-xxiv of the Course book
  3. 17th January 2012: Differential Equations: Who? What? Why? Where? How?

    Topics
    Second order linear differential equations
    Aims
    know what it means to solve an ODE
    have seen the method of reduction of order
    Lecture Notes
    beamer
    trans
    handout
    No significant annotations were made
    Resources
    Ch4.1 (pp xxxv–xlv) of the Course book; especially Ex 26
  4. 18th January 2012: ODEs: The Simplest Case

    Topics
    Second order linear differential equations with constant coefficients
    Aims
    have seen how to solve an ODE with constant coefficients
    have seen how to classify the solutions
    have seen what this means for the example of a pendulum
    Lecture Notes
    beamer
    trans
    handout
    annotations
    Resources
    Ch4.3 of the Course book
  5. 24th January 2012: Inhomogeneous ODEs

    Topics
    Second order inhomogeneous linear differential equations
    Linear independence and the Wronskian
    Aims
    learn about linear independence and the Wronskian
    know what an inhomogeneous ODE is
    know how to solve one by “intelligent guesswork”
    Lecture Notes
    beamer
    trans
    handout
    annotations
    Note
    We did not get through all of the material in this lecture
  6. 25th January 2012: Solving Inhomogeneous ODEs

    Topics
    Second order inhomogeneous linear differential equations
    Aims
    know what an inhomogeneous ODE is
    know how to solve one by “intelligent guesswork”
    know how to solve one by “variation of parameters”
    Lecture Notes
    beamer
    trans
    handout
    annotations
    Note
    We did not get through all of the material in this lecture
    The “test your understanding” of linear independence page can be viewed as a PDF.
  7. 31st January 2012: Resonance

    Topics
    Second order inhomogeneous linear differential equations
    Aims
    know how to solve an inhomogeneous ODE by “variation of parameters”
    have seen the example of “resonance”
    Lecture Notes
    beamer
    trans
    handout
    annotations
  8. 1st February 2012: Linear Systems

    Topics
    Introduction to linear systems
    Aims
    know what a linear system is
    know what it means to solve one
    have seen the idea of how to solve one
    Lecture Notes
    beamer
    trans
    handout
    annotations
  9. 7th February 2012: Matrices and Linear Systems

    Topics
    Introduction to linear systems
    Aims
    know the relationship between linear systems and matrices
    know how to solve a linear system using matrices
    know how to classify a linear system by the number of solutions
    Lecture Notes
    beamer
    trans
    handout
    annotations
  10. 8th February 2012: Explorations in Matrixland

    Topics
    Introduction to matrices
    Aims
    have seen how to do basic matrix manipulations
    have seen what composition of matrices means
    have seen how to compose matrices
    Lecture Notes
    beamer
    trans
    handout
    annotations
  11. 14th February 2012: Building Up and Tearing Down

    Topics
    Linear independence and span
    Aims
    know how to describe solution sets
    know what linear independence means
    know how to relate linear transformations to matrices
    Lecture Notes
    beamer
    trans
    handout
    annotations
  12. 15th February 2012: Spans and Dependence

    Topics
    Linear independence and span
    Aims
    know what is the span of a family of vectors
    know how to describe solution sets
    know what linear independence means
    Lecture Notes
    beamer
    trans
    handout
    annotations
  13. 21st February 2012: Invertibility

    Topics
    Linear independence and invertible matrices
    Aims
    know about linear independence
    know what it means for a matrix to be invertible
    know how to determine invertibility
    know how to find the inverse using Gaussian Elimination
    Lecture Notes
    beamer
    trans
    handout
    annotations
  14. 22nd February 2012: Muddiest Point

    Lecture Notes
    annotations
  15. 28th February 2012: Inverting Matrices

    Topics
    Invertible matrices and determinants
    Aims
    know how to characterise invertible matrices
    know how to invert a matrix using G E
    know how to compute determinants using G E
    know when to invert a matrix and when not to
    Lecture Notes
    beamer
    trans
    handout
    annotations
  16. 29th February 2012: Determinants and Gaussian Elimination

    Topics
    Invertible matrices and determinants
    Aims
    know how to compute determinants
    see how Gaussian Elimination answers every question
    Lecture Notes
    beamer
    trans
    handout
    annotations
  17. 6th March 2012: Vector Spaces

    Topics
    Vector spaces and subspaces of Euclidean space
    Aims
    have seen the definition of a vector space
    have seen the definition of a subspace
    have seen examples from matrices
    Lecture Notes
    beamer
    trans
    handout
    annotations
  18. 7th March 2012: Describing a Subspace

    Topics
    Vector spaces and subspaces of Euclidean space
    Aims
    have seen how to associate subspaces to a matrix
    have seen how to associate matrices to subspaces
    have seen how to use Gaussian elimination to find the best descriptions
    Lecture Notes
    beamer
    trans
    handout
    annotations
  19. 13th March 2012: Bases

    Topics
    Bases of subspaces
    Aims
    have seen the definition of a basis of a subspace
    have seen examples of such
    have seen the basic properties of bases
    have seen how to use Gaussian Elimination to find them
    Lecture Notes
    beamer
    trans
    handout
    annotations
  20. 14th March 2012: Eigenvalues

    Topics
    Eigenvalues and Eigenvectors
    Aims
    have seen the definition of an eigenvalue
    have seen examples of such
    have seen the basic properties of eigenvalues
    Lecture Notes
    beamer
    trans
    handout
    annotations
  21. 20th March 2012: Finding Eigenvalues

    Topics
    Eigenvalues and Eigenvectors
    Aims
    Have seen how to find eigenvalues (for small examples)
    Have seen examples using eigenvalues and eigenvectors
    Have seen how to decompose a vector into eigenvectors
    Lecture Notes
    beamer
    trans
    handout
    annotations
  22. 21st March 2012: Things to do with Eigenvectors

    Topics
    Eigenvalues and Eigenvectors
    Aims
    Seen how to find eigenvectors and eigenvalues
    Seen how to use eigenvectors to study differential equations
    Seen how to “change bases”
    Lecture Notes
    beamer
    trans
    handout
    annotations
  23. 27th March 2012: Special Eigenvalues

    Topics
    Eigenvalues and Eigenvectors
    Aims
    seen how to “change bases”
    seen how to use eigenvectors to study differential equations
    seen how to use eigenvectors to study Markov chains
    Lecture Notes
    beamer
    trans
    handout
    annotations
  24. 28th March 2012: Another Angle of Attack

    Topics
    Inner Products and Least Squares
    Aims
    seen how to use angles to find “closest points”
    seen the definition of the dot product and its relationship to angles
    seen how to find the “closest solution” to a linear system
    Lecture Notes
    beamer
    trans
    handout
    annotations?
  25. 11th April 2012: Orthogonality and Closest Points

    Topics
    Inner Products and Least Squares
    Aims
    see how to use least squares to find closest points
    see how to use orthogonality to build closest points
    Lecture Notes
    beamer
    trans
    handout
    annotations?
  26. 17th April 2012: The Gram-Schmidt Algorithm

    Topics
    Inner Products and Orthogonal Bases
    Aims
    have learnt how to find orthogonal families
    have learnt about decompositions and how they simplify problems
    have seen how to work with orthogonal decompositions
    Lecture Notes
    beamer
    trans
    handout
    annotations?
  27. 18th April 2012: Orthogonal Eigenvectors

    Topics
    Inner Products, Orthogonal Bases, and Eigenvectors
    Aims
    have seen the link between symmetric matrices and orthogonal eigenvectors
    have seen how this relates to functions
    Lecture Notes
    beamer
    trans
    handout
    annotations?
  28. 24th April 2012: Quadratic Forms

    Topics
    Quadratic Forms
    Aims
    have seen the classification of quadratic forms
    have seen how to relate them to conic sections
    Lecture Notes
    beamer
    trans
    handout
    annotations?

Note: Sometimes the software doesn’t register that a file has been uploaded. If you are expecting a file to be there and it is listed, try clicking on the question mark. If the file has been uploaded, that will download the file.

category: tma4115