Matrix multiplication applies a linear transformation - rotation, scaling, shearing, or projection - to a vector or another matrix. Each column of the result is where a basis vector lands after the transformation.

A 2×2 matrix transforms 2D space: rotating by 90°, scaling axes, or reflecting. Chaining transformations means multiplying matrices: (AB)x = A(Bx) - apply B first, then A. In neural networks, each linear layer is a matrix multiplication that projects the input into a new representation space. Dimensionality reduction happens when the output has fewer dimensions than the input (weight matrix has more columns than rows).