| Vectors in plane and space.
| Lines and planes.
| Vectror spaces: defntion, examles, subsaces, sums of subspaces.
| Basis and dimensions. Linar indpendence. Change of basis.
| Inner product spaces (I): definition, examples, computations, orthonormal basis, Schwarz inequalty, orthogonal complement.
| Inner product spaces (II): Gram-Schmidt ortogonalization process, Gram deteminants. Linear manifolds, distances.
| Linear maps (I): definition, kernel, image, injective and surjective maps.
| Linear maps (II): the matrix of a linear map.
| Eigenvectors and eigenvalues of operators (and associated matrix). Characteristic polynomial. Cayley-Hamilton thoerem. Diagonal form. Diagonaziabel operators.
| The Jordan canonical form for operators (and associated matrix). Jordan basis, the Jordan matrix.
| Functions of a matrix. The n-th power of a matrix. Elementary functions of a matrix.
| The adjoint operator. Definition, properties, examples.
| Special operators, Properties of eigenvalues and eigenvectors.
| Bilinear forms, quadratic forms. The associated matrix.
| Conics and quadrics. Reduction to a canonical form. Geometric properties.
