GEOMETRY AND LINEAR ALGEBRA - 20091

courses

ID:

20091

Dettaglio:

SSD: Geometry Duration: 48 CFU: 6

Located in:

DALMINE

Url:

Course Details:

BUILDING TECHNOLOGY ENGINEERING - 20-R-TE/PERCORSO COMUNE Year: 1

Year:

2026

Date/time interval

Secondo Semestre (23/02/2027 - 06/06/2027)

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.

For those who are not admitted to Part B, the grade is insufficient, without numerical grade. For those who are admitted to Part B, the grade is the sum of the grades of Parts A and B.

The professor may request a further oral interrogation, even after the end of the written exam, if needed to asses the effective acquisition of the knowledge explained in the educational goals of the teaching and to check whether the grade corresponds to reality.

In case the grade is 5/30 or less, the student will not be allowed to participate to the first following exam.

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.

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.