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Description
This programme demonstrates how a linear equation can be represented as a plane with a vector normal to it. Further examples using 2 and 3 equations are used to develop the consequences of this rep...resentation more fully.
Module code and title: M203, Introduction to pure mathematics M203; 11 01-05-1980 1980 00:25:00 + Show more... Martin Wright Norman Gowar; Robin Wilson BBC Open University Coefficient vector; Coset; Homogenous equation; Linear equations; Linearly independent; Solution set Algebra; Equations; Mathematics Norman Gowar describes the connection between a plane with a vector normal to it and an equation. The plane is called the solution The equation is a linear equation which can be expressed in a matrix form, and in turn as the dot product of two vectors. He now looks at the relationship between the equation and the plan in more detail, first of all by making the equation equal to nought. This means that the vector x must be perpendicular to the vector y. The set of vectors x that meet this condition, specify a plane through the origin at angles to the coefficient vector. This is called the homogeneous plane. Having found the solution plane for the homogenous equation Norman Gowar now returns to the original equation in which a.x =21 and finds the solution plane for this equation. This turns out to be parallel to the plane H, and the plane is shown to be a coset of the subspace H. Robin Wilson now discusses further equations. Using animations he looks at two planes represented by two equations. The solution set for the homogeneous pair of equations is seen to be line H, running through the origin. For the non-homogeneous solutions to the equations the solution set L is shown to be parallel to the set H and also a coset of this set. Robin now looks at three equations in three unknowns. The solution set in this case can be just a point. Norman Gowar discusses equations for which there appear to be no solution sets. For equation representing two planes this occurs when they have the same coefficient vectors, when the equations represent parallel planes. Similarly for three equations. Using animation Norman Gowar examines the effect of altering one of these planes, to see if this can provide a common solution. As them are still two parallel planes there is no point common to each plane. By altering the planes again so that there are three non-parallel planes, there is still an empty solution set. By moving one of the plane parallel to itself however a whole line of solution is obtained. Through looking at the equations in matrix form Norman Gowar shows how to distinguish between these possibilities. He then reviews the findings of the programme. 6HT/72907 00525_4291 840 no