There exist several fine books on vectors which achieve concision by only looking at vectors from a single point of view, be it that of algebra, analysis, physics or numerical analysis (see, for example, [18], [19], [23] and [28]). This book is written in the belief that it is helpful for the future mathematician to see all these points of view. It is based on those parts of the first and second year Cambridge courses which deal with vectors (omitting the material on multidimensional calculus and analysis) and contains roughly 60 to 70 hours of lectured material.

The first part of the book contains first year material and the second part contains second year material. Thus concepts reappear in increasingly sophisticated forms. In the first part of the book, the inner product starts as a tool in two and three dimensional geometry and is then extended to ℝn and later to ℂn. In the second part, it reappears as an object satisfying certain axioms. I expect my readers to read, or skip, rapidly through familiar material, only settling down to work when they reach new results. The index is provided mainly to help such readers who come upon an unfamiliar term which has been discussed earlier. Where the index gives a page number in a different font (like 389, rather than 389) this refers to an exercise. Sometimes I discuss the relation between the subject of the book and topics from other parts of mathematics.