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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View Full Document1.1 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 2dimensional 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 3dimensional 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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 Spring '03
 Andant
 Vector Space, σ, Euclidean space, Howard Staunton

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