Università Roma Tre

- Linear algebra: linear equations, matrices, Gauss-Jordan reduction, rank of a matrix, solutions of systems of linear equations, sum and product of matrices, invertible matrices and their construction, Rouchè-Capelli theorem.
- Square matrices and determinants: definition of determinant, properties of determinants, determinants and invertible matrices.
- Vector spaces: the example of geometric vectors, definition and examples of vector spaces, linearly independent vectors, independence, finitely generated vector space, basis, change of basis
- Scalar products: definition, euclidean spaces, examples of scalar products, perpendicularity, orthogonal basis, orthonormal basis and orthogonal matrices.
- Cartesian cordinates: coordinates on an affine euclidean space, fundamental metric properties, basic affine and euclidean geometry in dimension n.
- Plane and Space Geometry: isometries of the euclidean plane - points, straight lines, circles in the plane, angle between two straight lines, metric formulae for plane geometry, straight lines and planes in a space, equations of straight lines, planes, spheres, circles.
- Linear maps: Kernel and Image of a linear map, associated matrix after fixing the basis, linear operators, eigenvalues and eigenvectors of a linear operator, characteristic polynomial, search for eigenvalues and eigenvectors
- Quadratic equations: conics in the cartesian plane, conics and symmetric matrices, classification up to isometries, canonical form of a conic, metrical properties, quadrics in the space, type and canonical form.

Suggested texts: F. Flamini, A. Verra Matrici e Vettori. Carocci Editore.