Linear Algebra, Theory And Applications, 2012a

12.8. THE DETERMINANT AND VOLUME 303
12.8 The Determinant And Volume
The determinant is the essential algebraic tool which provides a way to give a unified treatment
of the concept of p dimensional volume of a parallelepiped in R M . Here is the definition
of what is meant by such a thing.
Definition 12.8.1 Let u 1 , ··· , u p be vectors in R M ,M ≥ p. The parallelepiped determined
by these vectors will be denoted by P (u 1 , ··· , u p ) anditisdefinedas
⎧
⎫
⎨ p∑
⎬
P (u 1 , ··· , u p ) ≡ s
⎩ j u j : s j ∈ [0, 1]
⎭ .
The volume of this parallelepiped is defined as
j=1
volume of P (u 1 , ··· , u p ) ≡ v (P (u 1 , ··· , u p )) ≡ (det (u i · u j )) 1/2 .
If the vectors are dependent, this definition will give the volume to be 0.
First lets observe the last assertion is true. Say u i = ∑ j≠i α ju j . Then the i th row is
a linear combination of the other rows and so from the properties of the determinant, the
determinant of this matrix is indeed zero as it should be.
A parallelepiped is a sort of a squashed box. Here is a picture which shows the relationship
between P (u 1 , ··· , u p−1 )andP (u 1 , ··· , u p ).
✻
N
✣
θ
u p
✸
P (u 1 , ··· , u p−1 )
✲
In a sense, we can define the volume any way we want but if it is to be reasonable, the
following relationship must hold. The appropriate definition of the volume of P (u 1 , ··· , u p )
in terms of P (u 1 , ··· , u p−1 )is
v (P (u 1 , ··· , u p )) = |u p ||cos (θ)| v (P (u 1 , ··· , u p−1 )) (12.11)
In the case where p =1, the parallelepiped P (v) consists of the single vector and the one
dimensional volume should be |v| = ( v T v ) 1/2
. Now having made this definition, I will show
that this is the appropriate definition of p dimensional volume for every p.
Definition 12.8.2 Let {u 1 , ··· , u p } be vectors. Then
⎛⎛
≡ det ⎜⎜
⎝⎝
v (P (u 1 , ··· , u p )) ≡
u T 1
u T 2
.
u T p
⎞
⎞
( )
⎟ u1 u 2 ··· u p ⎟
⎠
⎠
1/2