This textbook provides a full treatment of Linear Algebra devoted to undergraduate and graduate physics students. Although the mathematical level is similar to the corresponding mathematical textbooks in regard to definitions, propositions and proofs, it adopts a language and approach more attuned to the reader familiarity with physics lectures and physics textbooks. A distinctive feature is the emphasis placed on the significance of bases within a vector space. As a result, students gain a deeper understanding of how vector indices, despite their abundance, serve not as enemies but as friends since they give additional information about the mathematical objects being used, and facilitate access to tensor formalism. The book offers numerous worked examples and exercises with solution hints to deepen this knowledge.