18 1. From the Real Numbers to the Complex Numbers
introduced an object i whose square is -1. The first approach uses matrices of real
numbers, and it conveys significant geometric information. The second approach
fully justifies starting with (15) and (16) and it provides a quintessential example
of what mathematicians call a quotient space.
A matrix approach to C. The matrix definition of C uses two-by-two matrices
of real numbers and some of the ideas are crucial to subsequent developments. In
this approach we think of C as the set of two-by-two matrices of the form (18),
thereby presaging the Cauchy-Riemann equations which will appear throughout
the book. In some sense we identify a complex number with the operation of mul-
tiplication by that complex number. This approach is especially useful in complex
geometry.
We can regard a complex number as a special kind of linear transformation
of R2. A general linear transformation (x,y) (ax + cy,bx + dy) is given by a
two-by-two matrix M of real numbers:
(17) M =
a c
b d
.
A complex number will be a special kind of two-by-two matrix. Given a pair of
real numbers a,b and motivated by (13), we consider the mapping L :
R2

R2
defined by
L(x,y) = (ax - by,bx + ay).
The matrix representation (in the standard basis) of this linear mapping L is the
two-by-two matrix
(18)
a -b
b a
.
We say that a two-by-two matrix of real numbers satisfies the Cauchy-Riemann
equations if it has the form (18). A real linear transformation from R2 to itself whose
matrix representation satisfies (18) corresponds to a complex linear transformation
from C to itself, namely multiplication by a + ib.
In this approach we define a complex number to be a two-by-two matrix (of real
numbers) satisfying the Cauchy-Riemann equations. We add and multiply matrices
in the usual manner. We then have an additive identity 0, a multiplicative identity
1, an analogue of i, and inverses of nonzero elements, defined as follows:
(19) 0 =
0 0
0 0
,
(20) 1 =
1 0
0 1
,
(21) i =
0 -1
1 0
.
If a and b are not both 0, then
a2
+
b2
0. Hence in this case the matrix
(22)
a
a2+b2
b
a2+b2
-b
a2+b2
a
a2+b2
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