Math Visualization List
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
Geometric Interpretation of Cramer's Rule
Solve x·v₁ + y·v₂ = b.
Det(A) = 1.0
Det(A₁) = 1.0 (b replaces v₁)
Det(A₂) = 1.0 (b replaces v₂)
x = Det(A₁)/Det(A) = 1.0
y = Det(A₂)/Det(A) = 1.0
Det(A₁) = 1.0 (b replaces v₁)
Det(A₂) = 1.0 (b replaces v₂)
x = Det(A₁)/Det(A) = 1.0
y = Det(A₂)/Det(A) = 1.0
v₁ (x, y)
v₂ (x, y)
Target Vector b
Intuition: Since b = x·v₁ + y·v₂, when you replace v₁ with b, the new area is only affected by x·v₁ (because y·v₂ is parallel to v₂, contributing nothing to the area). Therefore, the ratio of the new area to the old area is exactly x.