Linear Algebra
Cheat Sheet

Matrices, vectors, and operations. From basics to advanced topics.

Vectors

Operations

u + v = (u₁+v₁, u₂+v₂, u₃+v₃)
c · v = (cv₁, cv₂, cv₃)
|v| = √(v₁² + v₂² + v₃²)

Dot Product

u · v = u₁v₁ + u₂v₂ + u₃v₃
u · v = |u||v|cos(θ)

Cross Product

u × v = (u₂v₃ - u₃v₂, 
         u₃v₁ - u₁v₃, 
         u₁v₂ - u₂v₁)

Matrices

Basic Operations

(A + B)ᵢⱼ = Aᵢⱼ + Bᵢⱼ
(cA)ᵢⱼ = cAᵢⱼ
(AB)ᵢⱼ = Σₖ AᵢₖBₖⱼ

Determinant (2×2)

det(A) = ad - bc
A = [[a, b], [c, d]]

Inverse (2×2)

A⁻¹ = (1/det(A)) × 
      [[d, -b], [-c, a]]

Eigenvalues & Eigenvectors

Definition

Av = λv
where λ is eigenvalue
and v is eigenvector

Characteristic Equation

det(A - λI) = 0

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