Fundamentals of Linear Algebra and Complex Numbers

📋 Course Outline

1. Complex Number Parts & Functions
2. Binary Operations & Properties
3. Relations & Image Types
4. Uniqueness of Identity & Inverses
5. Linear Subspaces & Conditions
6. Basis & Dimension Theory
7. Sum & Direct Sum of Spaces
8. Rank & Maximal Independent Subset
9. Linear Transformations & Isomorphisms
10. Dual Space & Dual Basis
11. Matrix Representation & Operations
12. Kernel, Image & Rank of Operators

📖 1. Complex Number Parts & Functions

🔑 Key Concepts & Definitions

• Complex Number: A number of the form $z = a + bi$, where $a, b \in \mathbb{R}$ and $i^2 = -1$.

• Real Part ($\Re(z)$): The component $a$ of the complex number $z = a + bi$.

• Imaginary Part ($\Im(z)$): The component $b$ of the complex number $z = a + bi$.

• Modulus ($|z|$): The distance of $z$ from the origin in the complex plane, defined as $|z| = \sqrt{a^2 + b^2}$.

• Argument ($\arg(z)$): The angle $\theta$ between the positive real axis and the line segment from the origin to $z$, typically in $[-\pi, \pi)$.

• Trigonometrical (Polar) Form: Representation of $z$ as $z = r(\cos \theta + i \sin \theta)$, where $r = |z|$.

📝 Essential Points

• The real part and imaginary part are extracted directly from the algebraic form: $z = a + bi$.

• The modulus relates to the magnitude of the complex number and is used in the polar form.