Unit 3 Summary

Linear Algebra for Machine Learning · Axiom Academy

Unit 3: Matrices & Linear Transformations In this unit, you've mastered the fundamental concepts and techniques of matrices and linear transformations. Matrices are not just arrays of numbers—they're powerful tools that represent data, transformations, and systems. Matrices are rectangular arrays organized in m rows and n columns Entry aᵢⱼ refers to row i, column j Dimension notation: m×n means m rows, n columns Addition: Add corresponding entries (requires same dimensions) Scalar multiplication: Multiply every entry by a constant Multiplication: (AB)ᵢⱼ = row i of A · column j of B Transpose: Swap rows and columns, (A+B)ᵀ = Aᵀ + Bᵀ, (AB)ᵀ = BᵀAᵀ Every matrix represents a linear transformation T(x) = Ax Linearity: T(αu + βv) = αT(u) + βT(v) Transformations can rotate, scale, shear, project, and reflect Composition of transformations is matrix multiplication Identity: I acts as multiplicative identity, AI = A Diagonal: Only diagonal entries are nonzero Symmetric: Aᵀ = A (mirrors across diagonal) Orthogonal: QᵀQ = I (preserves lengths and angles) Triangular: Upper or lower triangular form Single number encoding crucial information about a matrix 3×3: Cofactor expansion along any row or column det(A) = 0 means A is singular (not invertible) Only exists for square matrices with det(A) ≠ 0 2×2 formula: A⁻¹ = (1/det(A)) × [[d, -b], [-c, a]] Solves systems: Ax = b ⟹ x = A⁻¹b Rank, Null Space & Column Space Rank: Dimension of column space (number of linearly independent columns)

This is the written version of the interactive lesson above. See the full Linear Algebra for Machine Learning course.