A Bird’s-Eye View of Linear Algebra: Why Is Matrix Multiplication Like That?

Matrix multiplication is a foundational operation for the modern AI revolution because it is the primary way high-dimensional vectors are manipulated. Complex data like images, text, and speech are embedded into vector spaces, and machine learning models like neural networks use linear layers to transform these vectors. The article details the algebraic properties of matrix operations, noting that while addition is simple and commutative, standard matrix multiplication is more convoluted and not commutative. This specific definition of matrix multiplication is crucial because it directly represents how linear maps are composed and applied to vectors, making it a fundamental building block for AI.