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...

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.

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.

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.

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.

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!

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...

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...

Vectors are directed line segments. They are quite different from points, since they can be added, multiplied by a scalar or number, and subtracted. The arithmetic with vectors is very useful in Linear Algebra, and here we use it to describe various standard applications, including affine combinations of two vectors to describe all the points on a line segment or line determined by two vectors. We use vectors to prove that the diagonals of a parallelogram bisect each other, and that the...

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.

This is the fourth lecture in our series on Linear Algebra. Here we introduce area as a determinant, first in two dimensions, then in three. We give a pictorial definition using the affine grid plane, then also a purely algebraic approach using Grassmann's bi-vectors. A bi-vector is a two dimensional analog of a vector, and captures the physical notions of torque, angular momentum, and force on a charged particle in a magnetic field. We find the classical formulas for 2 and 3 dimensional...