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July 27, 2010AlgTop0: 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.

...more May 07, 2010WildLinAlg5: Change of coordinates and determinants ...more April 16, 2010WildLinAlg12: 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.

...more March 24, 2010WildLinAlg13: 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....more March 24, 2010WildLinAlg10: 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....more March 23, 2010WildLinAlg11: Applications of 3x3 matrices ...more March 17, 2010WildLinAlg8: Inverting 3x3 matrices This is the 8th lecture in this

series on Linear Algebra. Here we solve the most fundamental problem inthe subject in the 3x3 case---in such a way that extension to higherdimensions becomes almost obvious.What is the fundamental problem? It

is: How to invert a change of coordinates? Or in matrix terms: How tofind the inverse of a matrix?And the answer rests squarely on the wonderful function called thedeterminant. Be prepared for some algebra, but it is beautiful algebra!...more March 17, 2010WildLinAlg7: 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 andreflections.Finally we discuss the connections with calculus, introducing the idea

that the derivative is really a linear transformation....more March 16, 2010WildLinAlg9: Three dimensional affine geometry This is the ninth lecture of this course on

Linear Algebra. Here we give a gentle introduction to three dimensionalspace, starting with the analog of a grid plane built from a packing ofparallelopipeds in space.We discuss two different ways of drawing 3D objects in 2D, emphasizing

the importance of parallel projection. Some discussion of the nature ofspace and modern physics, then an introduction of affine space viacoordinates. The distinctions between points and vectors is important,and we talk also about lines and planes....more March 03, 2010WildLinAlg3: Center of mass and barycentric coordinates Vectors are used throughout engineering and physics because their arithmetic parallels the standard ways of combining force, velocity, acceleration, and other quantities. In this video we look at some examples of each of these, and introduce some games to help you get the feel for velocity anjd acceleration in a vector framework.

Then we discuss Archimedes' Principle of the Lever, and give a vector reformulation of it, which naturally leads to the definition of the center of mass of two or more objects. Barycenteric coordinates link center of mass to affine combinations of vectors.

...more

The podcast currently has 13 episodes available.