Download Chapter 10: Complex vector spaces Section 10.1: Complex numbers

Chapter 10: Complex vector spaces
Section 10.1: Complex numbers
Definition: A complex number is an ordered pair of real numbers, denoted (a, b) or a + bi, where i2 = −1. If z = a + bi then the real part of z is
a (Re(z) = a) and the imaginary part of z is b (Im(z) = b).
Note: We can think of complex numbers geometrically as a point or
vector in R2 where the real part is the x coordinate, and the imaginary part
is the y coordinate.
Definition: Two complex numbers a + bi, c + di are equal, written
a + bi = c + di if a = c and b = d.
Arithmetic with complex numbers:
Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
Subtraction: (a + bi) − (c + di) = (a − c) + (b − d)i
Scalar multiplication: k(a + bi) = (ka) + (kb)i
Multiplication: (a + bi)(c + di) = ac + bdi2 + adi + bci = (ac − bd) + (ad + bc)i
Example:
1
Section 10.2: Division of complex numbers
Definition: The complex conjugate of a complex number z = a + bi,
denoted z̄, is defined as
z̄ = a − bi
Geometrically, this is the reflection of z across the real axis.
Definition: The modulus, or absolute value, of a complex number z =
a + bi, denoted |z|, is defined as
√
|z| = a2 + b2
Theorem 10.2.1: For any complex number z,
zz̄ = |z|2
Theorem 10.2.2: (Revised) If z2 6= 0, then
z̄2
1
= z2−1 =
z2
|z2 |2
and
z1
z1 z̄2
z1 z̄2
=
=
z2
|z2 |2
z2 z̄2
Theorem 10.2.3: Properties of the conjugate
For any complex numbers z1 , z2 ,
1. z1 + z2 = z̄1 + z̄2
2. z1 − z2 = z̄1 − z̄2
3. z1 z2 = z̄1 z̄2
2
4. z1 /z2 = z̄1 /z̄2
5. z̄ = z
Example:
3
Section 10.3: Polar form of a complex number
Definition: If z = a + bi is a complex number, then the polar form of z
is written as
z = r cos θ + r sin θi = r(cos θ + i sin θ)
where r = |z| and θ, called the argument of z, is the angle form the positive
real axis to the vector z. If −π < θ ≤ π then θ is called the principal argument of z.
Note: The polar form is also called the CIS form
Arithmetic using polar forms
If z1 = r1 (cos θ1 + i sin θ1 ), z2 = r2 (cos θ2 + i sin θ2 ), then
z1 z2 = r1 r2 (cos(θ1 + θ2 ) + i sin(θ1 + θ2 ))
z1
r1
=
(cos(θ1 − θ2 ) + i sin(θ1 − θ2 ))
z2
r2
Example:
4
Powers of complex numbers using polar forms
If z = r(cos θ + i sin θ), then
z n = r n (cos(nθ) + i sin(nθ))
If r = 1 then
(cos θ + i sin θ)n = cos(nθ) + i sin(nθ), DeMoivre’s formula
and
z
1/n
=
√
n
θ 2kπ
+
r cos
n
n
Example:
5
+ i sin
θ 2kπ
+
n
n
Notation: We can also write complex numbers as exponentials:
z = r(cos θ + i sin θ)
= reiθ
Then z̄ = re−iθ and eiθ = e−iθ
Example:
6
Section 10.4: Complex vector spaces
Definition: A complex vector space is a vector space whose scalars can
be complex numbers. In a complex vector space, a vector w
~ is a linear combination
of the vectors ~v1 , . . . , ~vr if it can be written as
w
~ = k1~v1 + k2~v2 + · · · + kr~vr
where k1 , k2 , . . . , kr are complex numbers. The definitions of linear independence, spanning, basis, dimension and subspace are unchanged in complex
vector spaces.
Definition: The space of n-tuples of complex numbers, denoted Cn , has
vectors of the form
~u = (u1 , u2, . . . , un )
where u1 = a1 + b1 i, u2 = a2 + b2 i, . . ..
Note: The standard basis for Cn is the same as the standard basis for Rn .
Example:
7
Definition: If ~u = (u1 , u2, . . . , un ), ~v = (v1 , v2 , . . . , vn ) are vectors in Cn ,
then their complex Euclidean inner product is defined as
~u · ~v = u1 v̄1 + u2 v̄2 + · · · + un v̄n
where v̄j is the complex conjugate of vj .
Theorem 10.4.1: Properties of the complex inner product
If ~u, ~v , w
~ are vectors in Cn , k is any complex number, then
1. ~u · ~v = ~v · ~u
2. (~u + ~v ) · w
~ = ~u · w
~ + ~v · w
~
3. (k~u) · ~v = k(~u · ~v)
4. ~v · ~v ≥ 0, ~v · ~v = 0 if and only if ~v~0
Definition: Euclidean norm of the vector ~u = (u1 , u2 , . . . , un ) in Cn is
defined as
k~uk = (~u · ~u)1/2
p
|u1 |2 + |u2 |2 + · · · + |un |2
=
Definition: Euclidean distance between two vectors ~u, ~v in Cn is defined
as
d(~u, ~v) = k~u − ~v k
p
|u1 − v1 |2 + |u2 − v2 |2 + · · · + |un − vn |2
=
Example:
8