ID:
20091
Dettaglio:
SSD: Geometry
Duration: 48
CFU: 6
Located in:
DALMINE
Url:
BUILDING TECHNOLOGY ENGINEERING - 20-R-TE/PERCORSO COMUNE Year: 1
ELECTRONICS AND AUTOMATION ENGINEERING TECHNOLOGY - 96-270R/Percorso comune Year: 1
Year:
2025
Overview
Date/time interval
Secondo Semestre (23/02/2026 - 06/06/2026)
Syllabus
Course Objectives
The keyword of the course is "multidimensionality". After studying, in the course "Analisi 1", phenomena characterized by only one independent variable and only one dependent variable, the student will be introduced to more realistic situations, where the number of variables is greater than one. In this course he will study the linear case only, while the nonlinear one will be tackled in the course "Analisi 2". At the end of the course, the student will have the basic notions of complex numbers and linear algebra; he will be able to apply linear algebra to the solution of linear systems and to the study of geometry in three dimensions.
Course Prerequisites
Plane Euclidean geometry: in particular, triangle criteria for equality and similarity, Euclid and Pythagoras theorems, elementary properties of polygons and circles. One-to-one correspondence between real numbers and points on a line; intervals, half line; Cartesian plane; distance between two points in the plane. Elementary loci in the plane: line (parallelism and orthogonality conditions), circle, ellipse, parabola, and hyperbole. Powers with integer exponent, properties of powers; polynomials: divisibility, Ruffini rule, roots, factorization. Trigonometry: measure of angles in radiant, fundamental identities.
Teaching Methods
The teaching is composed by lectures (40 hours), exercises (12 hours), and tutoring (12 hours). In all three activities the student is encouraged to participate with suggestions and proposals. If remote or blended teaching is necessary, some changes will be introduced, also for the exams.
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 web 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.
Contents
1) Complex numbers
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.
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.
Online Resources
More information
In case the lessons would be taught online or in mixed mode, some modifications may occur with respect to what is indicated in the syllabus so that course and exams can be done in those cases.
Degrees
Degrees (2)
BUILDING TECHNOLOGY ENGINEERING - 20-R-TE
Bachelor's Degree
3 years
ELECTRONICS AND AUTOMATION ENGINEERING TECHNOLOGY - 96-270R
Bachelor's Degree
3 years
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People
People (2)
Componente del Collegio di Disciplina
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Other
Main module
GEOMETRY AND LINEAR ALGEBRA