This book contains a collection of exercises designed to support and complement the theoretical exposition of standard topics for a first-year course in linear algebra and geometry at the university level. In this context, this exercise book naturally complements the text: F. Bottacin, "Linear Algebra and Geometry," Esculapio, Bologna, which will be referred to for the necessary theoretical concepts. In presenting solutions to the proposed exercises, my primary goal is to teach how to approach and solve problems. For this reason, in many cases, theoretical considerations have been included, and different solution methods have been proposed for the same type of exercise. The reader will thus be able to evaluate, on a case-by-case basis, the advantages and disadvantages of each method