This programme tests to see if the matrices, representing the linear transformations describing the symmetry group of a triangle, themselves form a group. It then tests to see if all 2 x 2 matrices... form a group.
|Module code and title:
|M203, Introduction to pure mathematics
|First transmission date:
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|Norman Gowar; Robin Wilson
|BBC Open University
|Symmetries; Transformations; Triangles
|Algebra, Abstract; Matrices; Symmetric matrices
|Norman Gowar describes the symmetry group of a triangle. He argues that each of the six symmetries can be thought of as linear transformations of the plane, these in turn can be represented by matrices. These matrices may form a group. Robin Wilson works out the matrices for the symmetries of the triangle. He explains the importance of matrices for making calculations and for expressing the coordinates of the image vector. Again he works through some examples to demonstrate this. Norman Gowar argues that as symmetries can be combined so also matrices must have a corresponding method of combination. Working through an example he derives a law of combination for these matrices. Robin Wilson works through some examples to show this general rule. Norman Gowar argues that this last finding, combined with the other properties of matrices allows us to state that these matrices form a group. He extends this to ask if all 2 by 2 matrices form a group. He examines the group axioms. He works out that not all matrices have inverses. Robin Wilson works through some examples to find a test of whether or not a matrix has an inverse. He explains this test both geometrically and algebraically. Norman Gowar now poses the question, does the set of all invertible 2 by 2 matrices form a group? The closure axiom presents some difficulties, but the student is asked to sort these out after the programme.
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