7—Operators and Matrices 157do it with your left hand. You cannot move one of these and put it on top of the other (unless you
haveveryunusual joints). One is a mirror image of the other.
The equation (7.39) is a special case of a rule that you’ve probably encountered elsewhere. You
compute the determinant of a square array of numbers by some means such as expansion in minors or
Gauss reduction. Here I’ve defined the determinant geometrically, and it has no obvious relation the
traditional numeric definition. They are the same, and the reason for that comes by looking at how the
area (or volume) of a parallelogram depends on the vectors that make up its sides. The derivation is
slightly involved, but no one step in it is hard. Along the way you will encounter a new and important
function:Λ.
Start with the basis~e 1 ,~e 2 and call the output of the transformation~v 1 =f(~e 1 )and~v 2 =f(~e 2 ).
The final area is a function of these last two vectors, call itΛ
(
~v 1 ,~v 2
)
, and this function has two key
properties:
Λ(
~v,~v
)
= 0, and Λ
(
~v 1 ,α~v 2 +β~v 3
)
=αΛ
(
~v 1 ,~v 2
)
+βΛ
(
~v 1 ,~v 3
)
(7.40)
That the area vanishes if the two sides are the same is obvious. That the area is a linear function of
the vectors forming the two sides is not so obvious. (It is linear in both arguments.) Part of the proof
of linearity is easy:
Λ
(
~v 1 ,α~v 2 ) =αΛ
(
~v 1 ,~v 2 )
simply says that if one side of the parallelogram remains fixed and the other changes by some factor,
then the area changes by that same factor. For the other part,Λ
(
~v 1 ,~v 2 +~v 3
)
, start with a picture and
see if the area that this function represents is the same as the sum of the two areas made from the
vectors~v 1 &~v 2 and~v 1 &~v 3.
~v 1 &~v 2 form the area OCBA. ~v 1 &~v 3 form the area OCED.
O
A
B
C
D
E
F G
H
J
~v 1
~v 2
~v 3
HDF∼=JEG
HDO∼=JEC
so
area HJGF=area DEGF=area OCBA
area OCJH=area OCED
add these equations:
area OCGF=area OCBA+area OCEDThe last line is the statement that sum of the areas of the two parallelograms is the area of the
parallelogram formed using the sum of the two vectors:
Λ(
~v 1 ,~v 2 +~v 3
)
= Λ
(
~v 1 ,~v 2
)
+ Λ
(
~v 1 ,~v 3
)
This sort of functionΛ, generalized to three dimensions, is characterized by(1) Λ(
α~v 1 +β~v 1 ′,~v 2 ,~v 3
)
=αΛ
(
~v 1 ,~v 2 ,~v 3
)
+βΛ
(
~v 1 ′,~v 2 ,~v 3
)
(2) Λ
(
~v 1 ,~v 1 ,~v 3
)
= 0 (7.41)
It is linear in each variable, and it vanishes if any two arguments are equal. I’ve written it for three
dimensions, but inNdimensions you have the same equations withNarguments, and these properties
hold for all of them.