Math Visualization List
Calculus
First Derivative Second Derivative Integration
Determinant
2x2 & 3x3 Determinants Cramer's Rule Permutations & Inversions Permutation Definition & Expansion Properties of Determinants: Row Operations Minors & Cofactors Two-Row Laplace Expansion Vandermonde Determinant
Matrix Operations
Matrix Addition, Subtraction & Scalar Multiplication Matrix Multiplication & Power Transpose, Determinant & Adjoint Inverse Matrix & Matrix Equations Elementary Transformations, Echelon & Canonical Form
Probability & Statistics
Conditional Probability, Independence & Total Probability Discrete Probability Distributions PMF & CDF 2D Random Vectors Conditional Probability Distributions Bayes' Theorem Numerical Characteristics of Random Variables Higher Moments & Covariance Matrix Law of Iterated Expectations Independence, Mean Independence & Uncorrelated Continuous Statistical Distributions
Vector Operations
Vector Addition, Subtraction & Scalar Multiplication Linear Combination, Span & Basis Matrices as Transformations Composition of Transformations 3D Linear Transformations Determinant Inverse, Column Space & Rank Cross-Dimensional Transformations Vector Angle Cross Product Change of Basis Eigenvectors & Eigenvalues Function Vector Spaces Geometric Interpretation of Cramer's Rule Gravity Simulation

Elementary Transformations, Echelon Form, Canonical Form & Elementary Matrices

Current Matrix A
Elementary Matrix E for This Operation
Canonical Form (Same Rank Target)

Manual Elementary Row Operations

1) Swap R ↔ R
2) Row Scaling R ← ×R
3) Row Addition R ← R + ×R

Step Log

Basic Introduction

Elementary Transformations & Elementary Matrices

Each elementary row operation is equivalent to left multiplication by an elementary matrix E, i.e., A' = EA.

Echelon Form & Reduced Echelon Form

  • REF: Pivot positions shift right gradually, elements below pivots are 0.
  • RREF: Based on REF, pivots are 1 and all other elements in pivot columns are 0.

Rank & Canonical Form

Matrix rank is the number of pivots. The canonical form with the same rank as the original matrix can be written as [I_r 0; 0 0].