Linear Algebra and Geometry

[2 CFU] Vector spaces on a field. Numerical vector spaces, standard scalar product, free and applied geometric vectors. Linear dependence, generators, bases and dimension. Subspaces of a vector space;

[1 CFU] Matrices. The vector space of matrices on a field. Transposed matrix. Square matrices of various types: triangular, diagonal, symmetrical. Rank of a matrix. Product rows by columns. The determinant of a square matrix: definition and main properties. Calculation methods. Laplace, Binet and «Largest minor» theorems. Elementary operations on the rows (or columns) of a matrix. Triangulation methods. Matters of invertibility. Systems of linear equations. Compatibility, equivalent systems. Rouchè-Capelli and Cramer theorems. Methods of calculating the solutions of a compatible system. Parametric systems;

[1 CFU] Matrix associated with a linear map. 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;

[1 CFU] Geometry of the plane. Parametric and Cartesian representation of the line. Directional vector. Bundles of straight lines. Notes on related issues in the plane: parallelism and incidence between lines. Notes on Euclidean issues in the plane: angles, orthogonality and distance. Geometry in a 3-dimensional affine space. Parametric and Cartesian representation of the line and the plane. Directional vector of the line and normal vector of the plane. Bundles of plans. Notes on related issues in space: parallelism and incidence between lines, between planes, and between a straight line and a plane. Notes on Euclidean issues in space: orthogonality and distance between lines planes;

[1 CFU] Notes on conics: projective extension, classification, polarity.