A succinct—if somewhat reductive—description of linear algebra is that it is the study of *vector spaces* over a *field*, and the associated structure-preserving maps known as *linear transformations*. These concepts are by now so standard that they are practically fossilised, appearing unchanged in textbooks for the best part of a century.

While modern mathematics has moved to more abstract pastures, the theorems of linear algebra are behind a surprising number of world-changing technologies: from quantum computing and quantum information, through control and systems theory, to big data and machine learning. All rely on various kinds of *circuit diagrams*, eg electrical circuits, quantum circuits or signal flow graphs. Circuits are geometric/topological entities, but have a vital connection to (linear) algebra, where the calculations are usually carried out.

In this article, we cut out the middle man and rediscover linear algebra itself as an algebra of circuit diagrams. The result is called graphical linear algebra and, instead of using traditional definitions, we will draw lots of pictures. Mathematicians often get nervous when given pictures, but relax: these ones are rigorous enough to replace formulas.