Download Class Notes

NOTES FOR HOMEWORK SET #1 FOR MATH 552
Remarks about Numbers: Recall the following sets of numbers:
•
•
•
•
Natural numbers, N : = {1, 2, 3, . . . }.
Integers, Z : = {. . . , −3, −2, −1, 0, 1, 2, 3, . . . }.
n
Rational numbers, Q : = { m
| n, m ∈ Z, m 6= 0}.
Real numbers, R, numbers with a decimal expansion, points on
the number line.
Suppose S is a set of “numbers”. Then S should be equipped with two
operations, addition + and multiplication (denoted by justaposition).
S could have the following properties. (In the following suppose x, y, z
are any members of the set S, i.e. x, y, z ∈ S.)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
Closure property of addition: x + y ∈ S.
Commutative property of addition: x + y = y + x.
Associative property of addition: (x + y) + z = x + (y + z).
Additive identity: 0 ∈ S and x + 0 = x. (Not true for N.)
Additive inverses: x + (−x) = 0, where −x ∈ S. (Not true for
N.)
Closure property of multiplication: xy ∈ S.
Commutative property of multiplication: xy = yx.
Associative property of multiplication: (xy)z = x(yz).
Multiplicative identity: 1 ∈ S and x1 = x.
Multiplicative inverses: if x 6= 0 then xx−1 = 1, where x−1 ∈ S.
(Not true for N, or Z.)
Distributive property: x(y + z) = xy + xz.
Any other set S besides Q or R which has all these properties could
perhaps also be considered a set of “numbers”.
Suppose S : = R2×2 , the set of all 2 × 2 matrices with real entries. In
linear algebra we define the operations of addition and multiplication of
members of R2×2 . In linear algebra all of these properties are shown to
hold (the additive identity is 0 = ( 00 00 ) and the multiplicative identity
is 1 = ( 10 01 )) except (7) and (10). So we do not call members of R2×2
“numbers”.
Definition of the set of Complex Numbers:
¢
¡
∈ R2×2 | a, b ∈ R}.
C : = { ab −b
a
1
2
In order to justify our calling C a set of ¡numbers
¢ we need to verify
0 0 ) = a −b when a = b = 0 and
properties
(1),
(6),
(7),
and
(10).
(
0 0
b a
¡
¢
( 10 01 ) = ab −b
when
a
=
1
and
b
=
0.
The
other properties follow
a
2×2
immediately since they hold for R .
Verifying Property (1):
¡ a −b ¢ ¡ a′ −b′ ¢ ¡ a+a′
+ b′ a′ = b+b′
b a
Verifying Property (6):
¡ a −b ¢ ¡ a′ −b′ ¢ ¡ aa′ −bb′
= ba′ +ab′
b a
b′ a ′
Verifying Property (7):
¡ a −b ¢ ¡ a′ −b′ ¢
b a
¡ a′
b′
−b′
a′
b′
a′
¢ ¡ a −b ¢
b a
=
=
−b−b′
a+a′
−ab′ −ba′
−bb′ +aa′
¡ aa′ −bb′
¢
¢
=
=
³
³
a+a′ −(b+b′ )
b+b′ a+a′
∈ C.
aa′ −bb′ −(ab′ +ba′ )
ab′ +ba′ aa′ −bb′
−ab′ −ba′
−bb′ +aa′
¡ a′ a−b′ b −a′ b−b′ a ¢
b′ a+a′ b −b′ b+a′ a
ba′ +ab′
´
¢
´
∈ C.
= the above.
Verifying Property (10): A 2×2 matrix has¡a multiplicative
inverse
¢
2
2
a −b
if and only if its determinant is nonzero. det
a + b .¡ This¢
¡ a −bb¢ a =
0
0
can be zero if and only if a = b = 0. So if b a 6= ( 0 0 ) then ab −b
a
has a multiplicative inverse:
¡ a −b ¢−1
¡ a b¢
1
= a2 +b
2
−b a .
b a
Notations
for Complex
Numbers. Define i = ( 01 −1
0 ). Notice that
¢
¡ −1
0
2
i = ii = ¡ 0 −1¢ = −1. Every member z of C can be written in the
= a1 + bi. We define ℜ(z) = a and ℑ(z) = b. The
form z = ab −b
a
text uses Re(z) for ℜ(z) and Im(z) for ℑ(z). These are called the real
and imaginary parts of z. The subset {a1 ∈ C | a ∈ R} ⊂ C has all
the properties of the set R of real numbers. It is very inconvenient
to distinguish between R and {a1 ∈ C | a ∈ R}, or between a = a1
and a1. This is what we mean when we write R ⊂ C. Thus instead
of writing a1 + bi we invariably write simply a + bi. We will use this
notation from now
¡ on.
¢ Notice that the text uses the notation z = (a, b)
instead of z = ab −b
a , but consequently has to give a strange definition
of multiplication for such ordered pairs.
Additional Definitions
√
√ involving Complex Numbers. We also
define ¡|z| : =¢ det z = a2 + b2 , called the modulus of z, and z : =
a b
z T = −b
(z T denotes the transpose of the matrix z), called the
a
conjugate of z.