Linear Algebra Essentials

📋 Course Outline

1. Vector Operations
2. Matrix Types
3. Matrix Operations
4. Determinant Calculation
5. Linear Systems
6. Eigenvalues and Eigenvectors
7. Vector Space Properties
8. Subspaces and Basis
9. Linear Transformations

📖 1. Vector Operations

🔑 Key Concepts & Definitions

• Vector: An ordered set of numbers representing a point or direction in space, denoted as ( \mathbf{v} = (v_1, v_2, ..., v_n) ).
• Vector Addition: The operation of adding two vectors component-wise: ( \mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2, ..., u_n + v_n) ).
• Scalar Multiplication: Multiplying a vector by a scalar ( c ), scaling each component: ( c\mathbf{v} = (cv_1, cv_2, ..., cv_n) ).
• Dot Product: A scalar obtained from two vectors, calculated as ( \mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + ... + u_n v_n ), measuring similarity or projection.
• Magnitude (Norm): The length of a vector, given by ( |\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + ... + v_n^2} ).