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MAT/03 Geometry
Written exam (exercises and numerical problems, possibly multiple-choice) and oral exam.
The course aims to introduce and formalise the fundamental concepts of linear algebra and Euclidean geometry. In particular, it seeks to show how linear algebra can be used to redefine the main incidence properties of points, lines and planes, as well as the notions of distance between points and of angle and orthogonality between lines and planes.
Geometric and algebraic structures. Vector spaces. Equivalence relations and free vectors. Numerical vector spaces and the standard scalar product. Linear dependence, generators, bases and dimension. Subspaces. Grassmann’s theorem. Matrices. Transpose matrix. Rank of a matrix. Row-by-column product. The determinant of a square matrix. Methods of calculation. Laplace’s, Binet’s and the bordered minors theorems. Elementary row (or column) operations on a matrix. Triangulation methods. Invertibility questions. Systems of linear equations. Consistency, equivalent systems. The Rouché-Capelli and Cramer theorems. Computing the solutions of a consistent system. Parametric systems. Linear maps. Kernel and image. Monomorphisms, epimorphisms and isomorphisms. The coordinate isomorphism. Matrix associated with a linear map. Endomorphisms, eigenvalues, eigenvectors and eigenspaces. The characteristic polynomial. Algebraic and geometric multiplicity of an eigenvalue. Diagonalisation of an endomorphism and of a matrix. The Spectral Theorem. Bilinear maps and forms. Scalar products. Angles and distances. Euclidean vector spaces. Orthogonal matrices and orthonormal bases. Orthogonal diagonalisation. Affine spaces and subspaces. Plane geometry. Parametric and Cartesian representation of a line. Pencils of lines. Outline of affine and Euclidean questions in the plane. Solid geometry. Parametric and Cartesian representation of a line and of a plane. Pencils of planes. Outline of affine and Euclidean questions in space: parallelism, orthogonality and incidence between lines, between planes, and between a line and a plane. The common perpendicular problem. Projective and complex extension of affine/Euclidean space. Study of conics: double points, polarity, classification. Diameters, asymptotes, axes, centre, vertices and foci.
at the end of the course, students must demonstrate that they:
The assessment will consider the student’s command of the mathematical tools used, ability to present and correct use of language, as well as the ability to support a discussion with examples and counterexamples, the ability to apply the knowledge acquired to the solution of simple geometric problems and, finally, the ability to contextualise this knowledge in more applied settings.
