444 CHAPTER 20. THEORY OF DETERMINANTS∗
20.3 p Dimensional ParallelepipedsThe determinant is the essential algebraic tool which provides a way to give a unifiedtreatment of the concept of p dimensional volume of a parallelepiped in RM . Here is thedefinition of what is meant by such a thing.
Definition 20.3.1 Let u1, · · · ,up be vectors in RM,M ≥ p. The parallelepiped de-termined by these vectors will be denoted by P(u1, · · · ,up) and it is defined as
P(u1, · · · ,up)≡
{p
∑j=1
s ju j : s j ∈ [0,1]
}=UQ, Q = [0,1]p
where U =(u1 · · · up
).The volume of this parallelepiped is defined as
volume of P(u1, · · · ,up)≡ v(P(u1, · · · ,up))≡ (det(G))1/2 .
where Gi j = ui ·u j. This G = UTU is called the metric tensor. If the vectors ui aredependent, this definition will give the volume to be 0.
First lets observe the last assertion is true. Say ui = ∑ j ̸=i α ju j. Then the ith row of Gis a linear combination of the other rows using the scalars α j and so from the propertiesof the determinant, the determinant of this matrix is indeed zero as it should be. Indeed,ui ·uk = ∑ j ̸=i α ju j ·uk .
A parallelepiped is a sort of a squashed box. Here is a picture which shows
P
up
w
θ
P = P(u1, · · · ,up−1)
the relationship between P(u1, · · · ,up−1) andP(u1, · · · ,up). In a sense, we can define the vol-ume any way desired, but if it is to be reasonable,the following relationship must hold. The appro-priate definition of the volume of P(u1, · · · ,up) interms of P(u1, · · · ,up−1) is v(P(u1, · · · ,up)) =∣∣up ·w
∣∣v(P(u1, · · · ,up−1)) (20.13)
where w is any unit vector perpendicular to each ofu1, · · · ,up−1. Note
∣∣up ·w∣∣= ∣∣up
∣∣ |cosθ | from thegeometric meaning of the dot product. In the case where p = 1, the parallelepiped P(v)
consists of the single vector and the one dimensional volume should be |v|=(vTv
)1/2=
(v ·v)1/2. Now having made this definition, I will show that det(G)1/2 is the appropriatedefinition of v(P(u1, · · · ,up)) for every p.
As just pointed out, this is the only reasonable definition of volume in the case of onevector. The next theorem shows that it is the only reasonable definition of volume of aparallelepiped in the case of p vectors because 20.13 holds.
Theorem 20.3.2 If we desire 20.13 to hold for any w perpendicular to each ui,then we obtain the definition of 20.3.1 for v(P(u1, · · · ,up)) in terms of determinants.
Proof: So assume we want 20.13 to hold. Suppose the determinant formula holdsfor P(u1, · · · ,up−1). It is necessary to show that if w is a unit vector perpendicular toeach u1, · · · ,up−1 then
∣∣up ·w∣∣v(P(u1, · · · ,up−1)) reduces to det(G)1/2. By the Gram