The prior section shows how matrices represent linear maps. A good strategy, on seeing a new idea, is to explore how it interacts with some already-established ideas. In the first subsection we will ask how the representation of the sum of two maps ${\displaystyle f+g}$ is related to the representations of the two maps, and how the representation of a scalar product ${\displaystyle r\cdot h}$ of a map is related to the representation of that map. In later subsections we will see how to represent map composition and map inverse.