7—Operators and Matrices 1857.6 Areas, Volumes, Determinants
In the two-dimensional example of arrows in the plane, look what happens to areas when an operator acts. The
unit square with corners at the origin and(0,1),(1,1), 1 ,0)gets distorted into a parallelogram. The arrows from
the origin to every point in the square become arrows that fill out the parallelogram.
What is the area of this parallelogram?
I’ll ask a more general question. (It isn’t really, but it looks like it.) Start withany region in the plane,
and say it has areaA 1. The operator takes all the vectors ending in this area into some new area of a sizeA 2 ,
probably different from the original. What is the ratio of the new area to the old one?A 2 /A 1. How much does
this transformation stretch or squeeze the area? What isn’t instantly obvious is that this ratio of areas depends
on the operatoronly,and not on how you chose the initial region to be transformed. If you accept this for the
moment, then you see that the question in the previous paragraph, in which I started with the unit square and
asked for the area into which it transformed, is the same question as finding the ratio of the two more general
areas. (Or the ratio of two volumes in three dimensions.)
This ratio is called the determinant of the operator.
The first example is the simplest. Rotations in the plane,Rα. Because rotations leave area unchanged,
this determinant is one. For almost any other example you have to do some work. Use the component form to
do the computation. The basis vector~e 1 is transformed into the vectorf 11 ~e 1 +f 21 ~e 2 with a similar expression
for the image of~e 2. You can use the cross product to compute the area of the parallelogram that these define.
For another way, see problem 3. This is
(
f 11 ~e 1 +f 21 ~e 2
)
×
(
f 12 ~e 1 +f 22 ~e 2)
=
(
f 11 f 22 −f 21 f 12)
~e 3 (25)The product in parentheses is the determinant of the transformation.det(f) =f 11 f 22 −f 21 f 12 (26)What if I had picked a different basis, maybe even one that isn’t orthonormal? From the definition of the
determinant it is a property of the operator and not of the particular basis and components you use to describe it,