Overview

Every matrix studied so far has been a fixed grid of numbers acting on column vectors. This topic makes the leap to the object the unit has been building toward: the linear transformation — a structure-preserving map between vector spaces, of which "$A$ acting on $\mathbf{x}$" is only the coordinate-dependent shadow. Once transformations are abstract, two things become possible that couldn't be asked of a bare matrix: the same map can be represented by different matrices depending on which basis is chosen (opening the door to similarity and, eventually, diagonalisation), and the four fundamental subspaces of Topic 3 get renamed kernel and image, with a single theorem — rank-nullity — proved in full generality rather than merely observed as a dimension-counting pattern.