The book contains examples of the witty application of linear algebra in various fields of mathematics - mainly in combinatorics, geometry and theory of algorithms. Each section is devoted to one significant result, its motivation and evidence. Understanding requires only some familiarity with linear algebra. The book contains many recognized mathematical pearls, including Hamming codes, the matrix tree theorem, the Lovas boundary for Shannon capacity, and a counterexample to the Borsuk hypothesis. Less well-known, but no less remarkable results are presented; among them are a quick test of associativity, Steinitz's lemma on ordering vectors, the theorem on non-growing integer partitions, and the application of the outer product when considering pairs of sets. Relatively simple results from the first miniatures give rich material that makes the university course of linear algebra come to life in memory. More difficult sections can be used in the course of linear algebraic methods for graduate students.