Linear Algebra Fundamentals

2/5/2024• 3 min read

Introduction to linear algebra concepts and applications

mathematicslinear-algebravectorsmatrices

Linear Algebra Fundamentals

Linear algebra is a branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces.

Introduction

Linear algebra is fundamental to many areas of mathematics and computer science, including machine learning, computer graphics, and scientific computing.

Vectors

A vector is an ordered list of numbers, typically represented as a column or row.

Vector Operations

Addition: Component-wise addition

[a₁]   [b₁]   [a₁ + b₁]
[a₂] + [b₂] = [a₂ + b₂]
[a₃]   [b₃]   [a₃ + b₃]

Scalar Multiplication: Multiply each component by scalar

    [a₁]   [ka₁]
k × [a₂] = [ka₂]
    [a₃]   [ka₃]

Dot Product: Sum of products of corresponding components

a · b = a₁b₁ + a₂b₂ + a₃b₃

Cross Product: Vector perpendicular to both input vectors (3D)

Matrices

A matrix is a rectangular array of numbers arranged in rows and columns.

Matrix Operations

Addition: Element-wise addition (same dimensions)

Multiplication:

C = A × B
C[i][j] = Σ A[i][k] × B[k][j]

Transpose: Flip matrix over its diagonal

Aᵀ[i][j] = A[j][i]

Determinant: Scalar value for square matrices

  • 2×2: det(A) = ad - bc
  • 3×3: More complex calculation

Inverse: Matrix A⁻¹ such that A × A⁻¹ = I

Systems of Linear Equations

A system of linear equations can be represented as:

Ax = b

Where:

  • A is coefficient matrix
  • x is variable vector
  • b is constant vector

Solving Methods

  1. Gaussian Elimination: Row operations to row-echelon form
  2. Gauss-Jordan Elimination: Reduced row-echelon form
  3. Cramer's Rule: Using determinants
  4. Matrix Inversion: x = A⁻¹b

Vector Spaces

A vector space is a set of vectors closed under addition and scalar multiplication.

Properties

  • Closure under addition
  • Closure under scalar multiplication
  • Associativity and commutativity
  • Existence of zero vector
  • Existence of additive inverse

Subspaces

A subset of a vector space that is itself a vector space.

Eigenvalues and Eigenvectors

For matrix A, if:

Av = λv

Then:

  • λ is an eigenvalue
  • v is an eigenvector

Applications

  • Principal Component Analysis (PCA)
  • Google PageRank algorithm
  • Vibration analysis
  • Quantum mechanics

Applications in Computer Science

Computer Graphics

  • 3D transformations
  • Rotation matrices
  • Perspective projection

Machine Learning

  • Feature vectors
  • Weight matrices
  • Principal Component Analysis
  • Neural networks

Data Analysis

  • Dimensionality reduction
  • Clustering
  • Regression analysis

Important Theorems

  • Rank-Nullity Theorem: rank(A) + nullity(A) = n
  • Cayley-Hamilton Theorem: Matrix satisfies its characteristic equation
  • Singular Value Decomposition (SVD): A = UΣVᵀ

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