LINEAR ALGEBRA AND GEOMETRY P - Z
Academic Year 2023/2024 - Teacher: MONICA LA BARBIERAExpected Learning Outcomes
Knowledge: being able to compute the rank of a matrix, with or without a parameter, to study a vector space, to study a linear application, to determine eigenvalues and eigenspaces of an endomorphism, to diagonalize a matrix, to solve problems of linear geometry.
Understanding: fundamental definitions and theorems about vector spaces, linear applications and endomorphisms, constructions and theorems about lines and planes in the space.
Course Structure
Detailed Course Content
Linear Algebra:
1. General information on sets, operations. Applications between sets, image and counterimage, injectivity, surjectivity, bijective applications. Sets with operations, the main geometric structures: groups, rings, fields.
2. The vectors of ordinary space. Sum of vectors, product of a number and a vector. Scalar product, vector product, mixed product. Components of vectors and operations using components.
3. Vector spaces and their properties. Examples. Subspaces. Intersection, union and sum of subspaces. Linear independence, relative criterion. Generators of a space. Basis of a space, method of successive discards, completion to a basis. Steinitz lemma*, dimension of a vector space. Grassmann formula*. Direct sums.
4. General information on matrices. Rank. Reduced matrices and reduction method. Elementary matrices. Product of matrices. Linear systems, Rouché-Capelli theorem. Solving linear systems with the (Gaussian) reduction method, free unknowns. Inverse of a square matrix. Homogeneous systems and subspace of solutions.
5. Determinants and their properties. Laplace's theorems*. Calculation of the inverse of a square matrix. Binet's theorem*. Cramer's theorem. Kronecker theorem*.
6. Linear maps and their properties. Core and image of a linear application. Injectivity, surjectivity, isomorphisms. The space L(V,W), its isomorphism* with K^{m,n}. Study of linear applications. Change of base, similar matrices.
7. Eigenvalues, eigenvectors and eigenspaces of an endomorphism. Characteristic polynomial. Size of eigenspaces. Independence of the eigenvectors. Simple endomorphisms and diagonalization of matrices.
Geometry:
1. Linear geometry in the plane. Cartesian coordinates and homogeneous coordinates. Lines and their equations. Intersections between lines. Slope coefficient. Distances. Bundles of lines.
2. Linear geometry in space. Cartesian coordinates and homogeneous coordinates. Planes and their equations. The lines, their representation. Improper elements. Angular properties of lines and planes. Distances. Bundles of plans.
3. Coordinate changes in the plane, rotations and translations.
Textbook Information
1. S. Giuffrida, A. Ragusa: Corso di Algebra Lineare. Il Cigno Galileo Galilei, Roma, 1998.
2. G. Paxia, Lezioni di Geometria. Spazio Libri, Catania, 2000.
3. P. Bonacini, M. G. Cinquegrani, L. Marino. Algebra lineare: esercizi svolti. Cavallotto Edizioni, Catania, 2012.
4. P. Bonacini, M. G. Cinquegrani, L. Marino. Geometria analitica: esercizi svolti. Cavallotto Edizioni, Catania, 2012.
5. E. Sernesi. Geometria 1. Bollati Boringhieri, 2000.
More didactic material is available at https://studium.unict.it/ and https://algebralineare-geometria.webnode.it/
Course Planning
| Subjects | Text References | |
|---|---|---|
| 1 | Introduction to set theory. | Testo 3: cap 1,3 - Testo 1: cap. 1 |
| 2 | introduction to fields and vector spaces. | Testo 3: cap 1,3 - Testo 1: cap. 1 |
| 3 | Determinant of a matrix. Rank and reduction of a matrix. Resolution of a linear system. | Testo 3: cap 1,3 - Testo 1: cap. 1 |
| 4 | Vector spaces. Generators and linear independence. Subspaces. Base and components with respect to a base. Dimension of a vector space. | Testo 3: cap. 2 - Testo 1: cap. 2 |
| 5 | Sum and intersection of vector spaces. Extracting a base from a set of generators and expanding a linearly independent set to a base. | Testo 3: cap. 2 - Testo 1: cap. 2 |
| 6 | Linear applications and their assignment. Studying a linear application. Computation of images and inverse images. | Testo 3: cap. 4 - Testo 1: cap. 3,4,5 |
| 7 | Base change matrices and similar matrices. Operations with linear applications. | Testo 3: cap. 4 - Testo 1: cap. 3,4,5 |
| 8 | Eigenvalues, eigenvectors and eigenspaces. Characteristic polynomial. Algebraic and geometric multiplicity of an eigenvalue. Endomorphisms and diagonalization. | Testo 3: cap. 5 - Testo 1: cap. 6,7,8 |
| 9 | Applications under conditions. Restrictions and extensions of linear applications. | Testo 3: cap. 5 - Testo 1: cap. 6,7,8 |
| 10 | Generalities on vector calculus. Cartesian coordinates and homogeneous coordinates. Assignment of lines and planes and their equations. Points at infinity. Intersections. Parallelism and orthogonality. Pencils of lines and planes. Distances. . | Testo 4: cap. 1,2,3 - Testo 2: cap. 1 |