Geometry and Algebra - a.y. 2016/2017
Period (Year/Semester): I/I
Required/expected prior knowledge: Nothing
Francesca Cioffi - Classe A-E Fuorigrotta
Maria Rosaria Celentani - Classe F-R Fuorigrotta
Ulderico Dardano - Classe S-Z Fuorigrotta
Antonio Fontana - Classe A-I San Giovanni a Teduccio
Antonio Fontana - Classe J-Z San Giovanni a Teduccio
The student must know the definitions and statements set forth in the lessons and must be able to articulate the demonstrations of the main statements, must be able to apply the acquired theoretical tools. The student must also be able to generalize the techniques studied and solve problems and exercises.
Tables of contents:
Geometric and algebraic structures. Vector spaces. Equivalence relations and free vectors. Numerical vector spaces and standard scalar product. Linear dependence, generators, bases and dimension. Subspaces. Grassmann theorem. Matrices. The space of matrices over a field. Transposed matrix. Square matrices of various types: triangular, diagonal, symmetric. Rank of a matrix. Row by column product. The determinant of a square matrix. Calculus of the determinant. Laplace, Binet theorems. Elementary operations on the row (columns, respectively) of a matrix. Triangulation of a matrix. Invertibility. Systems of equations. Compatibility and equivalent systems. Cramer and Rouchè-Capelli theorems. The computation of solutions of a compatible system. Parametric systems. Linear applications. Kernel and Image. Monomorphisms, epimorphisms and isomorphisms. The coordinate isomorphism. The matrix associated with a linear application. Endomorphisms, eigenvalues, eigenvectors and eigenspaces. The characteristic polynomial. Algebraic and geometric multiplicity of an eigenvalue. Diagonalization of an endomorphism and a matrix. The Spectral Theorem. Euclidean vector spaces. Orthogonal matrices and orthonormal bases. Planar geometry. Parametric and Cartesian representation of a line. One-parameter families of lines. A primer on affine and Euclidean questions on planar geometry. The geometry of a 3-dimensional space. Parametric and Cartesian representation of lines and planes. One-parameter families of planes. A primer on affine and Euclidean questions on a (three dimensional) space: parallelism, orthogonality and incidence among lines and planes. The problem of the common perpendicular of two non parallel lines. Metric problem in the space.
Lessons and exercises.
Textbooks and learning aids:
Elementi di Geometria e Algebra lineare - F.Orecchia - Ed. Liguori.
Esercizi di Geometria I - F.Orecchia - Ed. Aracne.
Assessment will be written and oral. Question are: multiple choiche tests, open questions, numerical exercises