The teaching is organized as follows:

Learning objectives

The main notions and techniques of linear algebra and matrix theory are presented, focusing both on theoretical and computational aspects. Moreover, the course provides an introduction to planar and spatial geometry, within the projective, affine, and euclidean setting, both analytical (coordinates, matrices) and synthetic tools will be employed.
At the end of the course the student must be able to demonstrate an adequate synthesis and abstraction ability and be able to formalize and solve linear algebra and geometric problems.

Program

Preliminary remarks: Review of trigonometry: angles in degrees and radians, trigonometric functions and their geometric interpretation. Slope of a line and perpendicularity relations. Trigonometric equations. Formulas for addition, duplication, and bisection.

Complex numbers: definition and representation in the Argand–Gauss plane. Algebraic operations, conjugacy, polar form, and exponential form. Euler's formula and connections with trigonometric functions. De Moivre's formula, nth roots, and solving equations in the complex field.

Vector spaces: definition and main examples (geometric vectors, polynomials, functions). Vector subspaces, linear combinations, sets of generators. Linear dependence and independence. Bases, dimension, and coordinates of a vector with respect to a basis. Affine spaces and their interpretation as solutions to linear systems.

Linear systems and matrices: echelon reduction and the Gauss–Jordan algorithm. Matrix rank, interpretation by rows and columns. Rouché–Capelli theorem. Solution spaces, sum and intersection of subspaces. Grassmann's formula. Elementary matrices and their role in Gaussian transformations.

Determinants and invertibility: definition of determinant, fundamental properties, calculation by Laplace expansion. Relationship between determinant and rank. Matrix invertibility criteria. Matrix inverse by cofactor method and Gaussian reduction. Cramer's formula.

Linear applications: definition and examples. Kernel and image. Nullity theorem and rank. Isomorphisms and their characterization. Matrix representation of a linear application, associated matrices, and change of basis. Endomorphisms and similarity between matrices.

Eigenvalues and diagonalization: characteristic polynomial and its properties. Eigenvalues and eigenvectors. Eigenspaces, algebraic and geometric multiplicity. Diagonalizability criteria. Trace, determinant, and connections with eigenvalues.

Scalar products and Euclidean spaces: real and complex scalar products, matrix representation. Orthogonality and orthonormality. Orthogonal spaces. Fundamental inequalities. Gram–Schmidt orthogonalization and QR factorization. Orthogonal projections and least squares.

Orthogonal and spectral transformations: isometries, orthogonal and unitary matrices. Self-adjoint endomorphisms and the spectral theorem. Singular value decomposition (SVD) and applications to low-rank compression.

Geometric applications: rotations, reflections, and homotheties. Vector product and its geometric properties. Tangent line and normal plane to curves in space.

Learning assessment procedures

Evaluation is through a written test. The exam includes a qualifying test, designed to verify possession of basic skills. Passing the test is a prerequisite for taking the next test.