This is a two-semester text covering techniques in linear algebra and multivariable calculus which every practitioner of mathematical sciences ought to know. It is intended as an advanced course for degrees that require a strong mathematical background. In keeping with its subject, the book adopts a how to do it approach. However, a broad explanatory perspective is offered to underpin this approach and so to skirt the proverbial pitfall of the cook-book method – a student all stuffed up with recipes and not knowing where to go when faced with a nonstandard problem.

So the aim here is two-fold: to equip students with the routine skills necessary in most modern applications, and to help develop a firm understanding of the underlying mathematical principles. Emphasis has therefore been placed equally on heart-of-the-matter, down-to-earth explanations, and on the presentation of an abundant stock of worked examples and exercises.

As regards the explanatory material, the guiding principle has been to keep the argument informal but nonetheless careful and accurate. This does justice to the mathematics without becoming obsessive over the kind of detail that a fully rigorous approach would require. The explanations are always concrete and geometric in nature. In fact, the geometric approach naturally leads to the adoption of linear algebra, the language of vectors, as a unifying frame for both parts of the course. A topic like non-linear programming then comes across as a fine exemplar of the interplay of the two areas.