Requirements
- Lyceum
Features
- Mathematical analysis
Target audiences
- University students
Recall on algebraic structures: sets, operations and their properties. Groups, rings, fields. The field of rational numbers, the field of real numbers, the field of complex numbers. Examples of finite fields.
Vector spaces: definition and examples. Vector space Kn, on the field K. Linear combinations of vectors. Linearly independent vectors, linearly dependent vectors. Generator systems. Bases. The canonical basis of Kn. Components of a vector with respect to a fixed basis. Finitely generated vector spaces. The dimension of a vector space. Vector subspaces. Intersection of vector subspaces, sum of vector subspaces. Vector subspaces generated by any set of vectors. The Grassmann formula (dimension of the space sum of two vector subspaces).
Linear functions and matrices: definition and examples. The kernel and image of a linear function. Nullity and rank of a linear function. Isomorphisms of vector spaces. Matrices, sum of matrices, product between a real number and a matrix. The vector space of matrices with n lines and m columns. Linear functions and bases: the matrix associated with a linear function. Product (rows by columns) of a matrix times a vector. The composition of two linear functions. The product (rows by columns) of two matrices. The inverse of a linear function and the inverse of a matrix. Necessary and sufficient conditions for the invertibility of a matrix. The determinant of a square matrix. The main properties of the determinant. The determinant of the product of two matrices (Binet's Theorem), the determinant of the inverse of a matrix. Laplace's formula (development of a determinant by rows or columns). Explicit formula for the inverse of a matrix. Elementary operations on the rows (or columns) of a matrix, reduction of a matrix to a triangular shape (upper or lower). Reducing a matrix to ladder form. Application to rank calculation.
Systems of linear equations: Cramer's Theorem, Rouché-Capelli Theorem. Solving a system of linear equations using the Gaussian elimination method. Homogeneous and non-homogeneous linear systems, relations between the set of solutions of a linear system and the solution space of the associated homogeneous linear system.
Eigenvalues and eigenvectors: eigenvalues and eigenvectors of an endomorphism of a vector space, eigenvalues and eigenvectors of a square matrix. Autospaces. Characteristic polynomial and characteristic equation. Algebraic and geometric multiplicity of an eigenvalue. Diagonalizability criterion. Similar matrices, diagonalizable matrices.
Scalar products: the norm of a vector of Rn, the dot product of two vectors of Rn, the angle between two vectors (the Cauchy-Schwarz inequality). Orthogonality between vectors and between vector subspaces. Decomposition of a vector as a sum of orthogonal vectors. Orthogonal projections. Orthogonal bases and orthonormal bases. The Gram-Schmidt orthogonalization procedure. Symmetric bilinear forms and their main properties. Eigenvalues and eigenvectors of symmetric matrices: orthogonally diagonalizable matrices.
Affine spaces and Euclidean affine spaces: definition and examples. Affine subspaces: parametric equations and Cartesian equations. Incident, parallel, skewed subspaces. Orthogonal subspaces. Distances between affine subspaces. Lines and planes in three-dimensional real affine space. One-plane normal vector. Distance of a point from a line and of a point from a plane. Distance between two skewed lines. Bundles of plans. Angle between incident lines. Angle between a straight line and a plane. Corner between two floors. Some notes on circumferences, spheres, cones and cylinders.
Course Features
- Lectures 0
- Quizzes 0
- Duration 5 hours
- Skill level Intermediate
- Language Italian
- Students 10
- Assessments Yes