Bring the chapter together: recognise a linear map, find its kernel and image, count dimensions, write its matrix in a basis, change the basis, and decide invertibility. No worked example sits above the answers.

A mechanic listens to an engine, reads the gauges and opens the bonnet, and each tells part of the story. Kernel, image, matrix and rank are the gauges of a linear map.

Decide first what is asked. Linear: test sums, multiples and the zero vector. Kernel and image: for a matrix, the null space and the column space. Dimensions: rank–nullity. A matrix in bases: coordinates of the images of the basis, as columns. A new basis: $P^{-1}AP$. Invertibility: any one of the equivalent conditions.