Page 248 - Linear Algebra Done Right

Volume
n
{(y 1 ,...,y n ) ∈ R : x j <y j <x j + r for j = 1,...,n};
you should verify that when n = 2, this gives a square, and that when
n = 3, it gives a familiar three-dimensional cube. The volume of a cube 239
n
n
in R with side length r is defined to be r . To define the volume of
n
an arbitrary set Ω ⊂ R , the idea is to write Ω as a subset of a union of Readers familiar with
many small cubes, then add up the volumes of these small cubes. As outer measure will
we approximate Ω more accurately by unions (perhaps infinite unions) recognize that concept
of small cubes, we get a better estimate of volume Ω. here.
Rather than take the trouble to make precise this definition of vol-
ume, we will work only with an intuitive notion of volume. Our purpose
in this book is to understand linear algebra, whereas notions of volume
belong to analysis (though as we will soon see, volume is intimately con-
nected with determinants). Thus for the rest of this section we will rely
on intuitive notions of volume rather than on a rigorous development,
though we shall maintain our usual rigor in the linear algebra parts
of what follows. Everything said here about volume will be correct—
the intuitive reasons given here can be converted into formally correct
proofs using the machinery of analysis.
n
For T ∈L(V) and Ω ⊂ R , define T(Ω) by
T(Ω) ={Tx : x ∈ Ω}.

Our goal is to find a formula for the volume of T(Ω) in terms of T
and the volume of Ω. First let’s consider a simple example. Suppose
n
λ 1 ,...,λ n are positive numbers. Define T ∈L(R ) by T(x 1 ,...,x n ) =
(λ 1 x 1 ,...,λ n x n ).If Ω is a cube in R n with side length r, then T(Ω)
n
is a box in R with sides of length λ 1 r,...,λ n r. This box has volume
n
n
λ 1 ...λ n r , whereas the cube Ω has volume r . Thus this particular T,
when applied to a cube, multiplies volumes by a factor of λ 1 ...λ n ,
which happens to equal det T.
As above, assume that λ 1 ,...,λ n are positive numbers. Now sup-
pose that (e 1 ,...,e n ) is an orthonormal basis of R n and T is the op-
erator on R n that satisfies Te j = λ j e j for j = 1,...,n. In the special
n
case where (e 1 ,...,e n ) is the standard basis of R , this operator is the
same one as defined in the paragraph above. Even for an arbitrary or-
thonormal basis (e 1 ,...,e n ), this operator has the same behavior as
the one in the paragraph above—it multiplies the j th basis vector by
a factor of λ j . Thus we can reasonably assume that this operator also
multiplies volumes by a factor of λ 1 ...λ n , which again equals det T.