03-28-03

03-28-03 - Lecture 21 Andrei Antonenko 1 General properties...

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Lecture 21 Andrei Antonenko March 28, 2003 1 General properties of “area”, “volume” and their gen- eralizations In this lecture we will give the general deﬁnition of the determinant of any square matrix. On the last lecture we introduced the oriented area of the parallelogram. The main prop- erties we used were the following: 1. area( a,b ) = - area( b,a ) for any vectors a and b . From this condition it follows that area( a,a ) = 0 for any vector a . We’ll prove it. Proof. Let area( a,b ) = - area( b,a ) for all vectors a and b . Let b = a . Then area( a,a ) = - area( a,a ) and so area( a,a ) = 0. 2. area( a 1 + a 2 ,b ) = area( a 1 ,b ) + area( a 2 ,b ) area( a,b 1 + b 2 ) = area( a,b 1 ) + area( a,b 2 ) . 3. area( e 1 ,e 2 ) = 1. From these properties we deﬁned the formula for the oriented area of the parallelogram. We should mention, that in order to derive this formula we needed ONLY these properties and nothing else! By the same method we can deﬁne the oriented volume of the parallelepiped. It will satisfy the similar properties, and we can get similar formula for it, using ONLY these properties. We will generalize this construction. 1

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1.1 Generalization of the properties The area was the function of 2 vectors in 2 dimensional space, and it was equal to the oriented area of the parallelogram on the 2-dimensional plane. Now we will consider the functions which takes n vectors from n -dimensional space as a parameters, and return a real number (For example, for oriented area we have n = 2, for oriented volume we have n = 3 — the function “volume” takes 3 vectors from the 3-dimensional space and returns a number — the value of the oriented volume). So, let f be a function with n parameters, each of them is a vector from n -dimensional space.
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03-28-03 - Lecture 21 Andrei Antonenko 1 General properties...

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