7—Operators and Matrices 186so you must get the same answer. But will I get the same simple formula ( 26 ) for the answer if I pick a different
basis? Nowthat’sa legitimate question. The answer is yes, but it takes some work to show it. What is the
determinant of Eq. ( 11 )?
The determinant can be either positive or negative. That tells you more than simply how the transformation
alters the area; it tells you whether it changes theorientationof the area. If you place a counterclockwise loop
in the original area, does it remain counterclockwise in the image or is it reversed? In three dimensions, the
corresponding plus or minus sign for the determinant says if you’re changing from a right-handed coordinate
system (the common one) to a left-handed one in which the order of the axesx-y-zis reversed.
det> 0det< 07.7 Matrices as Operators
There’s an important example of a vector space that I’ve avoided mentioning up to now. Example 5 in section
6.3is the set of n-tuples of numbers:(a 1 ,a 2 ,...,an). I can turn this on its side, call it a column matrix, and it
forms a perfectly good vector space. The functions (operators) on this vector space are the matrices themselves.
When you have a system of linear equations, you can translate this into the language of vectors.
ax+by=e and cx+dy=f −→(
a b
c d)(
x
y)
=
(
e
f)
Solving forxandyis inverting a matrix.
There’s an aspect of this that may strike you as odd. This matrix is an operator on the vector space of
column matrices. What are the components of this operator? What? Isn’t the matrix a set of components
already? That depends on your choice of basis. Take an example
M=
(
1 2
3 4
)
with basis ~e 1 =(
1
0
)
, ~e 2 =