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The first four axioms mean that V is an abelian group under addition. Elements of a vector space may have various nature; for example, they can be sequences, functions, polynomials or matrices.
Linear algebra is concerned with properties common to all vector spaces.
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This note covers the following topics: Linear Algebra, Matrix Algebra, Homogeneous Systems and Vector Subspaces, Basic Notions, Determinants and Eigenvalues, Diagonalization, The Exponential of a Matrix, Applications, Real Symmetric Matrices, Classification of Conics and Quadrics, Conics and the Method of Lagrange Multipliers, Normal Modes. This is a basic subject on matrix theory and linear algebra.