1. Basic Mathematical Concepts

Before diving into linear algebra, let’s first review some essential concepts: sets, mappings, and functions.

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Set

A set is a collection of objects that satisfy certain conditions.
Each object belonging to a set is called an element.

Examples:

• The set of natural numbers 1, 2, 3 → {1, 2, 3}

• The set of odd numbers → {1, 3, 5, 7, …}

Ways to Represent a Set

• Roster Form
  List out each element of the set.

  • Example: A = {1, 2, 3, 4}

• Set-builder Form
  Specify the property that the elements must satisfy.

  • Example: B = { x | x is a natural number less than or equal to 10 }

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Basic Concepts

• Subset
  If every element of set A is also in set B, then A is a subset of B.

  • Notation: A ⊆ B

  • Example: {1, 2} ⊆ {1, 2, 3}

• Empty Set
  A set with no elements.

  • Notation: ∅ or {}

  • Example: {x | x is an integer greater than 5 and less than 4} = ∅

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Set Operations

• Union
  The set containing all elements from both sets.

  • Notation: A ∪ B

  • Example: {1, 2} ∪ {2, 3} = {1, 2, 3}

• Intersection
  The set of elements common to both sets.

  • Notation: A ∩ B

  • Example: {1, 2} ∩ {2, 3} = {2}

• Difference
  The set of elements in one set but not in the other.

  • Notation: A – B

  • Example: {1, 2, 3} – {2, 3} = {1}

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Function / Mapping

A mapping (or function) assigns each element of a set A to exactly one element of a set B.

• A: Domain

• B: Codomain

• The actual set of mapped values in B: Range (Image)

Example:
f: A → B, f(x) = x²

• A = {1, 2, 3} (domain)

• B = {1, 4, 9, 16, 25} (codomain)

• Range = {1, 4, 9}

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Types of Functions

• Surjective (Onto)
  The range is equal to the codomain.
  → Every element in the codomain is mapped.

• Injective (One-to-one)
  Different inputs always map to different outputs.
  → If x₁ ≠ x₂, then f(x₁) ≠ f(x₂).

• Bijective (One-to-one Correspondence)
  Both injective and surjective.
  → A perfect one-to-one mapping between the domain and codomain.

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Inverse Function

If a function f: A → B is bijective, then there exists an inverse mapping g: B → A.
This is called the inverse function of f and is denoted by f⁻¹.

Example:

• f(x) = 2x (domain = all real numbers, codomain = all real numbers)

• Inverse: f⁻¹(x) = x/2

Matrix

A matrix is an arrangement of numbers or symbols in a rectangular form.
Matrices are mainly used in linear algebra to represent data or linear transformations.

Example:

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Basic Terminology

• Element / Entry
  Each individual number contained in a matrix.

• Square Matrix
  A matrix in which the number of rows equals the number of columns.

• Main Diagonal
  The diagonal running from the top-left to the bottom-right of a square matrix.

• Diagonal Matrix
  A square matrix in which all the elements outside the main diagonal are zero.

  

• Identity Matrix (Unit Matrix)
  A square matrix whose main diagonal elements are all 1 and whose other elements are all 0.

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Vector

A vector refers to a matrix that has only a single row or a single column.

• A vector consisting of a single row is called a row vector.

• A vector consisting of a single column is called a column vector.

2. Applications of Linear Algebra

Linear algebra is not just about learning how to manipulate matrices. In reality, it serves as the fundamental language of computer science, data science, and engineering. Let’s take a look at some representative applications.

• Data Representation and Processing
  Vectors and matrices provide the basic framework for storing and operating on structured data. For instance, in machine learning, input datasets are typically represented as matrices.

• Webpage Ranking
  Google’s PageRank algorithm models the web’s link structure as a matrix and evaluates the importance of each page through eigenvector computations.

• Computer Graphics
  3D modeling and transformations such as rotation, translation, and scaling are all expressed as matrix multiplications. Modern graphics in games and movies rely heavily on linear algebra.

• Robotics
  The position and orientation of robotic arms or drones are handled through coordinate transformations, which are calculated using matrix operations.

• Electrical Circuit Analysis
  The equations governing a circuit’s nodes form systems of linear equations, which can be expressed and solved using matrices.

• Fourier Transform
  Decomposing signals into frequency components via the Fourier transform can be viewed as a linear operation, and it is fundamental in image and audio processing.

• Dimensionality Reduction (PCA)
  Principal Component Analysis (PCA) reduces unnecessary dimensions in data analysis, relying on eigenvalue decomposition.

• Multivariate Gaussian Distribution
  The covariance matrix captures correlations between variables, enabling the modeling of multidimensional probability distributions—widely used in machine learning and statistics.

• Kalman Filter
  An algorithm that filters noise and estimates hidden states from sensor data. It is based on matrix representations of linear dynamic systems.