Mathematics 185 – Intro to Complex Analysis
Fall 2009 – M. Christ
Solutions to Selecta from Problem Set 1
1.1.10(a)
Let
z
∈
C
. Define
φ
z
:
C
→
C
by
φ
z
(
w
) =
zw
. Regard
φ
z
as a linear transfor
mation from
R
2
to
R
2
. Calculate its matrix representation, with respect to the standard
basis of
R
2
.
Solution.
First of all, since
φ
z
(
w
+ ˜
w
) =
φ
z
(
w
) +
φ
z
( ˜
w
) and
φ
z
(
tw
) =
tφ
z
(
w
) for any
w,
˜
w
∈
C
and
t
∈
R
,
φ
z
is indeed a linear transformation of
R
2
, if we identify complex
numbers with elements of
R
2
in the standard way.
I will systematically write
u
+
iv
= (
u, v
) to denote the correspondence between elements
of
C
, and elements of
R
2
.
Then
φ
z
(1
,
0) =
z
·
(1 +
i
0) =
z
=
x
+
iy
= (
x, y
).
Similarly
φ
z
(0
,
1) =
z
·
(0 +
i
) =
iz
=
ix

y
= (
y, x
). Thus the matrix representation of
φ
z
, with
respect to the standard basis, is
x

y
y
x
; this is the unique matrix
A
which satisfies
A
1
0
=
x
y
and
A
0
1
=

y
x
.
1.1.12(b)
Using only the axioms for a field, prove that
1
z
1
+
1
z
2
=
z
1
+
z
2
z
1
z
2
for any two nonzero
complex numbers.
Solution.
This is first of all an exercise in critical reading.
We are asked to prove
something involving
1
w
for various complex numbers
w
.
This cannot possibly be done,
using only the axioms for a field — because those axioms say nothing about
1
w
. We have to
supplement the axioms with the notational definition
1
w
=
w

1
. And we need to recognize
that
z
1
+
z
2
z
1
z
2
means (
z
1
+
z
2
)(
z

1
1
z

1
2
).
I will also use part (a) of this problem, in the corresponding form (
z
1
z
2
)

1
=
z

1
1
z

1
2
.
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 Spring '07
 Lim
 Math, Addition, Complex number, 1 1 z2

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