Learning objectives
The aim of this course is to provide students with essential tools in Linear Algebra and in Geometry; students are required also to apply their knowledge and understanding to concrete problems.
Prerequisites
Trigonometry, solving linear and quadratic equations, solving linear systems. Definitions of functions, injectivity and surjectivity.
Course unit content
This course is an introduction to different aspects of Linear Algebra and Geometry.
The first part is devoted to vectors, matrices and linear systems. Subsequently, we will study vector spaces, linear maps and the diagonalization of linear operators.
Full programme
0. Preliminaries: Real and complex numbers
1. Vectors in R^n and C^n. Operations on vectors and properties (scalar and Hermitian products, distances, angles, vector product,.)
2. Matrices and their properties. Determinant: Laplace expansion and basic properties. Binet theorem. Row and column elementary operations on matrices. Computation of the inverse matrix. Rank of a matrix. Symmetric and Hermitian matrices.
3. Linear systems: Gauss-Jordan method and Rouché Capelli theorem.
4. Real and complex vector spaces. Linear combinations of vectors: linear dependence/independence. Generators, bases and dimension of a vector space.
5. Linear maps. Definition of kernel and image; fundamental theorem on linear maps. Matrix representation of a linear map and change of bases. Isomorphisms and inverse maps.
5. Endomorphisms of a vector space: eigenvalues, eigenvectors and eigenspaces. Characteristic polynomial. Algebraic and geometric multiplicity. Diagonalizable endomorphisms.
6. Complements of algebra/geometry.
Bibliography
Alessandrini, L., Nicolodi, L., Geometria e algebra lineare, con esercizi svolti, Ed. Uninova(PR) 2012.
Silva A., Algebra Lineare, Edizioni Nuova Cultura, 1994
Abate, M., Geometria. McGraw-Hill.
Notes by the teacher.
Teaching methods
In the lectures we shall propose formal definitions and proofs, with significant examples and applications, and several exercises. Exercises are an essential tool in Linear Algebra; they will be proposed also in addition to lectures, in a guided manner.
Teaching materials will be eventually uploaded on Elly.
Assessment methods and criteria
The exam will be both written and oral. There might be intermediate tests during the course in replacement of the final written exam.
Other information
Lecture attendance is strongly recommended.
2030 agenda goals for sustainable development
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