Complex Vector Space: Linear Methods of AI - Artificial Intelligence & Data Science (Bachelor)

Introduction to Complex Numbers

Before understanding complex vector spaces, we need to understand complex numbers first. Imagine we have an imaginary unit $i$ with the special property $i^2 = -1$ . Complex numbers are numbers that consist of a real part and an imaginary part.

The set of complex numbers is defined as $\mathbb{C} := \{z = x + iy : x, y \in \mathbb{R}\}$ . In every complex number $z = x + iy$ , the part $x = \text{Re}(z)$ is called the real part and the part $y = \text{Im}(z)$ is called the imaginary part.

Using the property $i^2 = -1$ , we can perform arithmetic operations on complex numbers such as addition, subtraction, multiplication, and division. The set $(\mathbb{C}, +, \cdot)$ forms a field.

Conjugate and Modulus

For a complex number $z = x + iy$ , the complex conjugate is defined as $\overline{z} = x - iy$ . This conjugate is useful in various calculations.

The modulus or absolute value of a complex number is calculated with the formula:

|z| = \sqrt{z \cdot \overline{z}} = \sqrt{x^2 + y^2}

This modulus gives the "distance" of the complex number from the origin in the complex plane.

Fundamental Theorem of Algebra

The fundamental theorem of algebra is an important result that distinguishes complex polynomials from real polynomials. This theorem states that every non-constant polynomial with complex coefficients must have a complex root.

Now, let's look at a very important result. Every non-constant polynomial

p(z) = a_n z^n + a_{n-1} z^{n-1} + \cdots + a_1 z + a_0, \quad z \in \mathbb{C}

with degree $n \geq 1$ and complex coefficients $a_k \in \mathbb{C}$ for $k = 0, 1, \ldots, n$ with $a_n \neq 0$ , has at least one complex root. This means there exists a number $z_* \in \mathbb{C}$ such that $p(z_*) = 0$ .

This theorem is very important because it guarantees that in the field of complex numbers, every polynomial equation always has a solution.

Definition and Axioms of Complex Vector Space

After understanding complex numbers, we can extend the concept of vector spaces from real scalars to complex scalars. Vector spaces can not only be defined with scalars from $\mathbb{R}$ , but also from other fields such as $\mathbb{C}$ .

A set $V$ with an addition operation

+ : V \times V \to V : (x, y) \mapsto x + y

and a scalar multiplication operation

\cdot : \mathbb{C} \times V \to V : (\lambda, x) \mapsto \lambda \cdot x

is called a complex vector space if it satisfies the following axioms:

Vector Addition Axioms

Associative: $(x + y) + z = x + (y + z)$ for all $x, y, z \in V$
Commutative: $x + y = y + x$ for all $x, y \in V$
Identity Element: There exists an element $0 \in V$ such that $x + 0 = x = 0 + x$ for all $x \in V$
Inverse Element: For every $x \in V$ there exists an element $-x \in V$ such that $x + (-x) = 0 = (-x) + x$

Scalar Multiplication Axioms

Associative multiplication: $(\lambda \mu) \cdot x = \lambda \cdot (\mu \cdot x)$ for all $\lambda, \mu \in \mathbb{C}, x \in V$
Unit element: $1 \cdot x = x$ for all $x \in V$

Distributive Axioms

Distributive over vector addition: $\lambda \cdot (x + y) = \lambda \cdot x + \lambda \cdot y$ for all $\lambda \in \mathbb{C}, x, y \in V$
Distributive over scalar addition: $(\lambda + \mu) \cdot x = \lambda \cdot x + \mu \cdot x$ for all $\lambda, \mu \in \mathbb{C}, x \in V$

Elements $x \in V$ are called vectors, and elements $\lambda \in \mathbb{C}$ are called scalars.

Differences from Real Vector Spaces

It is important to understand that in complex vector spaces, we do not use special properties of real numbers such as ordering or real analysis. All results that apply to vector spaces remain valid for complex vector spaces, especially those related to matrices, systems of linear equations, and determinants.

The main difference lies in the scalar field used. Real vector spaces use $\mathbb{R}$ as the scalar field, whereas complex vector spaces use $\mathbb{C}$ which provides more flexibility in calculations.