The chapter introduces the concept of linear representations of finite groups, where a group homomorphism from a finite group \( G \) to the group of invertible linear transformations on a finite-dimensional vector space \( V \) over a field \( K \) is called a linear \( K \)-representation of \( G \). The study of representations provides insights into the structure of \( G \). Key applications include Burnside's theorem on solvability of groups with specific orders, Feit-Thompson's theorem on solvability of groups of odd order, and the classification of finite simple groups. The chapter also discusses the associated character, which is derived from the representation, and the group algebra \( KG \), which allows the study of representations through module theory. The text covers the construction of new modules from existing ones using tensor products and contragredients, and explores the relationship between representations and modules through category theory. Additionally, it delves into matrix representations, submodules, and orthogonality relations, leading to Schur's Lemma and the orthogonality of irreducible characters. The chapter concludes with a discussion on the number of simple modules and further orthogonality relations, providing a comprehensive overview of the theory of linear representations of finite groups.The chapter introduces the concept of linear representations of finite groups, where a group homomorphism from a finite group \( G \) to the group of invertible linear transformations on a finite-dimensional vector space \( V \) over a field \( K \) is called a linear \( K \)-representation of \( G \). The study of representations provides insights into the structure of \( G \). Key applications include Burnside's theorem on solvability of groups with specific orders, Feit-Thompson's theorem on solvability of groups of odd order, and the classification of finite simple groups. The chapter also discusses the associated character, which is derived from the representation, and the group algebra \( KG \), which allows the study of representations through module theory. The text covers the construction of new modules from existing ones using tensor products and contragredients, and explores the relationship between representations and modules through category theory. Additionally, it delves into matrix representations, submodules, and orthogonality relations, leading to Schur's Lemma and the orthogonality of irreducible characters. The chapter concludes with a discussion on the number of simple modules and further orthogonality relations, providing a comprehensive overview of the theory of linear representations of finite groups.