Linear Transformations

A matrix doesn't just sit there — it acts. Multiply Ax and you've moved a vector to a new place: stretched, rotated, reflected, sheared. The transformations that behave 'nicely' — preserving straight lines through the origin and keeping the grid parallel — are called linear, and the deep theorem is that every linear transformation IS a matrix. Graphics, animation, and neural-network layers are all linear transformations applied to vectors.