Matrices as Linear Transformations

Linear Algebra for Machine Learning · Axiom Academy

What is a Linear Transformation? A linear transformation is a function T that maps vectors from one space to another, preserving the operations of vector addition and scalar multiplication. A transformation T is linear if for all vectors u, v and scalars α, β: Matrix Representation of Transformations To find the matrix of a transformation T, apply T to the standard basis vectors. The results become the columns of A. 1. Rotation (counterclockwise by angle θ) Scale by factor c in x-direction, d in y-direction: Shear in x-direction with parameter k: Composition of Transformations When you apply one transformation, then another, you're composing transformations. The combined transformation is matrix multiplication! Example: Rotate by 45°, then scale by 2: Key Properties of Linear Transformations Preserve lines: Lines map to lines (or points) Preserve collinearity: Collinear points remain collinear Associative composition: T₃(T₂(T₁(x))) = (T₃∘T₂∘T₁)(x) = (A₃A₂A₁)x

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