In the course students are taught the basics of linear algebra and analytic geometry in the plane and in space. In particular the essential notions for solving a system of linear equations are developed, to calculate the rank of a matrix and of its other invariants. As far as the notions of analytical geometry are concerned, particular attention will be paid to the notion of scalar product and to the study of conics and quadrics
teacher profile teaching materials
- 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.
Further information on other useful texts for consultation and performance of exercises will be provided at the beginning of the course.
Some handouts will be distributed.
Programme
- 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.
Core Documentation
Matrices and Vectors' by F. Flamini and A. Verra (Carocci Editore).Further information on other useful texts for consultation and performance of exercises will be provided at the beginning of the course.
Some handouts will be distributed.
Type of evaluation
written and oral examination