knowledge/docs/linear_algebra.txt

LINEAR ALGEBRA FORMULAS AND CONCEPTS

Matrices:
- Addition: [A + B]ᵢⱼ = aᵢⱼ + bᵢⱼ
- Scalar multiplication: [kA]ᵢⱼ = k × aᵢⱼ
- Matrix multiplication: [AB]ᵢⱼ = Σₖ aᵢₖbₖⱼ
- Transpose: [Aᵀ]ᵢⱼ = aⱼᵢ

Properties:
- (AB)ᵀ = BᵀAᵀ
- (AB)C = A(BC)
- A(B+C) = AB + AC
- AB ≠ BA in general

Determinants:
- 2×2: det|a b; c d| = ad - bc
- 3×3: Use cofactor expansion
- det(AB) = det(A) × det(B)
- det(Aᵀ) = det(A)

Inverse:
- AA⁻¹ = A⁻¹A = I
- (AB)⁻¹ = B⁻¹A⁻¹
- (Aᵀ)⁻¹ = (A⁻¹)ᵀ
- A⁻¹ exists ⟺ det(A) ≠ 0

System of Equations:
- AX = B has unique solution if det(A) ≠ 0
- X = A⁻¹B
- Cramer's rule: xᵢ = det(Aᵢ)/det(A)

Common Mistakes:
- Matrix multiplication is not commutative
- Check dimensions before multiplying
- Determinant of product ≠ product of determinants