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episodes

  • AlgTop0: Introduction to Algebraic Topology

    This is the Introductory lecture to a beginner's course in Algebraic Topology, MATH5665, given by N J Wildberger of the School of Mathematics and Statistics at UNSW in 2010. The course is suitable for 3rd and 4th year mathematics majors, hopefully with some prior knowledge of group theory. Others with some mathematical maturity should be able to follow along with most of the material, which is highly visual.

    This first lecture introduces some of the main topics of the course and presents three problems to get students thinking topologically. Topics include: curves, winding number and curvature, two dimensional topological spaces and classification of surfaces using Conway's ZIP proof, polyhedra and Euler number, vector fields, the fundamental group, three dimensional manifolds and quaternions, and homology.

    I suggest what might be the two most important and interesting objects in the history of mathematics (you can see if you agree!) I also give three problems. One is a paper cutting exercise, another is Sam Loyd's pencil trick, and the third is a popular puzzle involving a block of wood, a loop of string and two balls.

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  • WildLinAlg12: Generalized dilations and eigenvectors

    By studying how Bob would view a dilation in Rachel's framework, we are led to the notion of a generalized dilation. Going from one basis to the other involves a 'Change of basis matrix'.

    We show how these ideas lead naturally to the important concepts of eigenvectors and associated eigenvalues, and give examples of finding them.

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  • WildLinAlg10: Equations of lines and planes in 3D
    Lines in the plane can be characterized by either parametric or Cartesian equations. The space of all such lines is naturally a Mobius band. Lines and planes in 3 dimensional space are then studied and drawn, including both Cartesian and parametric equations.
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  • WildLinAlg13: Solving a system of linear equations
    Row reduction or Gaussian elimination solves a system of linear equations in stages, by continually combining the equations to successively simplify the system by eliminating variables. We frame the algorithm using the augemented matrix of the system, performing elementary row operations. The first aim is to reduce the matrix to an equivalent one in row echelon form.
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  • WildLinAlg7: More applications of 2x2 matrices

    We continue discussing 2x2 matrices, their interpretation as

    linear transformations of the plane, how to analyse rotations,
    including a rational formulation, and how to combine rotations and
    reflections.

    Finally we discuss the connections with calculus, introducing the idea

    that the derivative is really a linear transformation.

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  • WildLinAlg8: Inverting 3x3 matrices

    This is the 8th lecture in this

    series on Linear Algebra. Here we solve the most fundamental problem in
    the subject in the 3x3 case---in such a way that extension to higher
    dimensions becomes almost obvious.

    What is the fundamental problem? It

    is: How to invert a change of coordinates? Or in matrix terms: How to
    find the inverse of a matrix?
    And the answer rests squarely on the wonderful function called the
    determinant. Be prepared for some algebra, but it is beautiful algebra!

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  • WildLinAlg9: Three dimensional affine geometry

    This is the ninth lecture of this course on

    Linear Algebra. Here we give a gentle introduction to three dimensional
    space, starting with the analog of a grid plane built from a packing of
    parallelopipeds in space.

    We discuss two different ways of drawing 3D objects in 2D, emphasizing

    the importance of parallel projection. Some discussion of the nature of
    space and modern physics, then an introduction of affine space via
    coordinates. The distinctions between points and vectors is important,
    and we talk also about lines and planes.

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  • WildLinAlg1: Introduction to Linear Algebra
    Linear Algebra is the study of affine geometry and its transformations. In this lecture we set up the basic idea of affine geometry in a visual way and introduce the main problem of the subject: how to go from one affine coordinate system to another.
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