DALMINE
Overview
Date/time interval
Syllabus
Course Objectives
Course Prerequisites
Teaching Methods
Assessment Methods
The exam aims to verify the achievement by students of the educational objectives described above. In particular: - Mastery of the methods and techniques developed - Awareness of their theoretical foundations - Appropriateness of the language used. The exam can be taken only by students who have fulfilled their OFA in mathematics. It is divided into two parts: Part A and Part B. Part A is a multiple choice test consisting of 10 (theoretical or practical) questions. Every correct answer is awarded 1 point, each incorrect answer is given -1/3. A minimum threshold, to be admitted to Part B, could be fixed. In case, this will be pointed out on the course e-learning page. Part B requires the resolution of some exercises and the exposition of some theoretical arguments. The score takes into account the correctness, clarity and the ability to justify the conclusions. The final grade is the sum of the marks obtained in Part A and Part B. The commission also reserves the right to re-examine any student after the correction of the written tests, if it deems it necessary to acquire further evaluation elements.
Contents
Sum, product, conjugate, modulus, inverse, and quotient. Representation in the plane. Real and imaginary part. Trigonometric form. N-th power and N-th root of a complex number. Fundamental theorem of algebra.
2) Vectors and matrices
The space R^n and its operations: sum, product by a scalar, scalar product. Matrices. Operations on matrices: sum, product by a scalar, product of matrices. Symmetric, triangular and diagonal matrices. Determinant. Geometrical meaning of the determinant. Inverse matrix. Three-dimensional vectors: vector product, mixed product and their geometrical meaning. Characteristic (or rank) of a matrix. Kronecker method.
3) Geometry in 3D space
Parametric representation of a straight line. Planes: parametric and cartesian equations. Cartesian representation of a straight line. Parallelism and orthogonality relations.
4) Vector subspaces of R^n and linear maps
Linear combinations. Linear dependence and independence. Vector subspaces. Bases and dimension. Subspace generated by a finite number of vectors. Orthonormal bases in R^n and orthogonal matrices. Linear maps from R^m to R^n. Linear map associated with a matrix. Kernel and image of a linear map. Rank–nullity theorem. Representative matrix of a linear map from R^n to R^n. Change of basis.
5) Linear systems
Cramer's theorem. Homogeneous systems. Rouché-Capelli theorem. Gauss method.
6) Diagonalizable matrices. Eigenvectors and eigenvalues. Quadratic forms. Conics and quadrics.