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episodes

  • WildLinAlg2: Geometry with vectors
    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 medians of a triangle all meet.
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  • WildLinAlg3: 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.

     

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  • WildLinAlg4: Area and volume

    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 determinants.

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