Applied Mathematics Essentials

Applied math provides the computational and analytical tools underpinning science and engineering. Core areas: LinAlg, calculus, DiffEq, probability/stats, and numerical methods.

## Linear Algebra

Vectors: dot product a·b = |a||b|cosθ = Σaᵢbᵢ (scalar, measures projection/similarity). Cross product a×b = |a||b|sinθ n̂ (vector, perpendicular to both, magnitude = parallelogram area). Orthogonal vectors: a·b = 0.

Matrices: m×n array, multiplication (AB)ᵢⱼ = Σₖ aᵢₖbₖⱼ. Not commutative: AB ≠ BA generally. Identity matrix I: AI = IA = A. Inverse A⁻¹ exists iff det(A) ≠ 0. For 2×2: A⁻¹ = (1/det)[d,-b;-c,a]. Transpose: (AB)ᵀ = BᵀAᵀ. Orthogonal matrix: QᵀQ = I (columns are orthonormal).

Determinant: det(A) = ad-bc for 2×2. Properties: det(AB) = det(A)det(B), det(Aᵀ) = det(A), row swap changes sign, det = 0 iff singular. Geometric interpretation: absolute value = volume scaling factor of linear transformation.

Eigenvalues/vectors: Av = λv, find λ from det(A-λI) = 0 (characteristic polynomial). Eigenvectors span eigenspaces. Symmetric matrices: real eigenvalues, orthogonal eigenvectors. Spectral decomposition A = QΛQᵀ. Applications: PCA (variance maximization), vibration modes, stability analysis (eigenvalues of Jacobian).