[Axler] Complex Numbers and Vector Spaces

Linear Algebra: Complex Numbers and Vector Spaces

Based on Linear Algebra Done Right (Section: $\mathbf{R}, \mathbf{C}$, and $\mathbf{F}^n$)

While we might initially focus on real numbers, complex numbers become necessary because real numbers alone are insufficient for solving all polynomial equations (e.g., $x^2 + 1 = 0$ has no real solution). To resolve this, we introduce the imaginary unit $i$, defined such that $i^2 = -1$.

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1. Complex Numbers ($\mathbf{C}$)

Definition

A complex number is an ordered pair of real numbers, denoted as $a + bi$, where:

• $a$ is the real part.
• $b$ is the imaginary part.
• The set of all complex numbers is denoted by $\mathbf{C}$.

Operations

• Addition: Add real and imaginary parts separately.
  $(a + bi) + (c + di) = (a + c) + (b + d)i$
• Multiplication: Use the distributive property and $i^2 = -1$.
  $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

Properties

Addition and multiplication in $\mathbf{C}$ satisfy standard algebraic properties (Field Axioms):

1. Commutativity: $\alpha + \beta = \beta + \alpha$ and $\alpha\beta = \beta\alpha$.
2. Associativity: $(\alpha + \beta) + \gamma = \alpha + (\beta + \gamma)$ and $(\alpha\beta)\gamma = \alpha(\beta\gamma)$.
3. Identities: $0$ is the additive identity; $1$ is the multiplicative identity.
4. Additive Inverse: For every $\alpha$, there exists $-\alpha$ such that $\alpha + (-\alpha) = 0$.
5. Multiplicative Inverse: For every $\alpha \ne 0$, there exists $\alpha^{-1}$ such that $\alpha \cdot \alpha^{-1} = 1$.
6. Distributivity: $\lambda(\alpha + \beta) = \lambda\alpha + \lambda\beta$.

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2. The Field $\mathbf{F}$

Throughout linear algebra, $\mathbf{F}$ denotes either the field of real numbers ($\mathbf{R}$) or the field of complex numbers ($\mathbf{C}$). Elements of $\mathbf{F}$ are called scalars.

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3. Vector Spaces: $\mathbf{F}^n$

We generalize the familiar concepts of $\mathbf{R}^2$ (plane) and $\mathbf{R}^3$ (space) to lists of length $n$.

Definition of $\mathbf{F}^n$

$\mathbf{F}^n$ is the set of all lists of length $n$ consisting of elements from $\mathbf{F}$.

$$ (x_1, x_2, \dots, x_n) $$

Example: $\mathbf{C}^4$ is the set of all lists of four complex numbers: $(z_1, z_2, z_3, z_4)$.

Operations on $\mathbf{F}^n$

• Addition: Add corresponding coordinates.
  $(x_1, \dots, x_n) + (y_1, \dots, y_n) = (x_1 + y_1, \dots, x_n + y_n)$
• Scalar Multiplication: Multiply each coordinate by the scalar $\lambda \in \mathbf{F}$.
  $\lambda(x_1, \dots, x_n) = (\lambda x_1, \dots, \lambda x_n)$

Notation:
We use a single bold letter (e.g., $\mathbf{x}$) to represent a vector.

• $\mathbf{x} = (x_1, \dots, x_n)$
• Zero vector: $\mathbf{0} = (0, \dots, 0)$

Geometric Interpretation

$\mathbf{F}^n$ is a powerful algebraic construct. While we can visualize $\mathbf{R}^2$ and $\mathbf{R}^3$, spaces like $\mathbf{R}^{100}$ or $\mathbf{C}^{400}$ rely on algebraic manipulation rather than geometric intuition. The ease of defining these spaces algebraically is a fundamental reason for the success of linear algebra.