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.
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
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Resources
Complex numbers
pp ix–xiv of the Course book
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
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Resources
Complex numbers
pp ix–xviii,xxiii-xxiv of the Course book
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
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No significant annotations were made
Resources
Ch4.1 (pp xxxv–xlv) of the Course book ; especially Ex 26
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
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Resources
Ch4.3 of the Course book
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
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Note
We did not get through all of the material in this lecture
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
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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 .
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”
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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
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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
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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
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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
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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
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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
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22nd February 2012: Muddiest Point
Lecture Notes
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Revised on August 16, 2013 22:50:49
by
Andrew Stacey
(80.203.115.236)