Fields. Equivalence relations. Matrices. Linear systems, Gauss' method, structure theorem. Vector spaces, subspaces, generators, linear independence, bases, sum of subspaces, Grassmann's formula, linear maps, kernel and image, isomorphisms. matrices associated to linear maps, the space of the linear maps between two vector spaces. Determinant and rank. Cramer's theorem. Rouché-Capelli theorem. Eigenvalues and eigenvectors, characteristic polynomial, algebraic multiplicity, geometric multiplicity, diagonalizability, necessary and sufficient criterion for diagonalizability. Bilinear forms, associated matrices, Gram-Schmidt theorems, signature. Hermitian matrices, normal matrices, unitary matrices, hermitian forms. Spectral theorems. Vector product. Affine and metric geometry, parallelism, orthogonality, lines and planes in the space, conics.