LINEAR ALGEBRA AND GEOMETRY P - Z

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, to classify conics and quadrics and to study conics bundles in the plane.

Understanding: fundamental definitions and theorems about vector spaces, linear applications and endomorphisms, constructions and theorems about lines and planes in the space and conics in the plane, definitions and theorems about the classifications of quadrics.

Linear Algebra:

Generalities on set theory and operations. Maps between sets, image and inverse image, injective and surjective maps, bijective maps. Sets with operation, gropus, rings, fields.

Vectors in the ordinary space. Sum of vectors, product of a number and a vector. Scalar product, vector product. Components of vectors and operations with components.

Complex numbers, operations and properties. Algebraic and trigonometric form of complex numbers. De Moivre formula. nth root of complex numbers.

Vector spaces and properties. Examples. Subspaces. Intersection, union and sum of subspaces. Linear independence. Generators. Base of a vector space, completion of a base. Steinitz Lemma*, dimension of a vector space. Grassmann formula*. Direct sum.

Generalities on matrices. Rank. Reduced matrix and reduction of a matrix. Elementary matrices. Product of matrices. Linear systems. Rouchè-Capelli theorem. Solutions of linear systems. Homogeneous systems and space of solutions.

Determinants and properties. Laplace theorems*. Inverse of a square matrix. Binet theorem*. Cramer thoerem. Kronecker theorem*.

Linear maps and properties. Kernel and image. Injective and surjcetive maps. Isomorphisms. L(V,W) and isomomorphism with k^{m,n}. Study of a linear map. Base change.

Eigenvalues, eigenvectors and eigenspaces of an endomorphism. Characteristic polynomial. Dimension of eigenspaces. Independence of eigenvectors. Simple endomorphisms and diagonalization of matrices.

Geometry

Linear geometry on the plane. Cartesian coordinates and homogeneous coordinates. Lines and their equations. Intersection of lines. Angular coefficient. Distances. Pencils of lines.

Linear geometry in the space. Cartesian coordinates and homogeneous coordinates. Planes and their equation. Lines and their representation. Ideal elements. Angular properties of lines and planes. Distances. Pencils of planes.

Change of coordinates in the plane, rotations and translations. Conics and associated matrices, orthogonal invariants. Reduced equations, reduction of a conic in canonic form. Classification of irreducible conics. Study of equations in canonic form. Circle. Tangent lines. Pencils of conics.

Quadrics in the space and associated matrices. Irreducible quadrics. Vertices and dengerate quadrics. Cones and cylinders. Reduced equations, reduction in canonic form. Classification of non degenerate quadrics. Sections of quadrics with lines and planes. Lines and tangent planes.

The proofs of the theorem signed with * can be ometted.

1. S. Lang, Introduction to linear algebra. Springer; 2nd edition, 1985.

Linear Algebra exercises

1. Determine the kernel and the image of a given linear map depending on a parameter.
2. Given an endomorphism depending on a parameter, determine whether it is diagonalizable. When possible, give a basis of the space formed by eigenvectors of the endomorphism.
3. Solution of a system of linear equations depending on a parameter. Given an endomorphism, compute the image and the pre-image of a vector and/or a vector (sub)space.
4. Exercises on vector spaces: dimension and direct sum. Induced linear maps.

Esercizi di Geometria

1. Linear geometry on the plane: parallel and orthogonal lines, distances, orthogonal projections, angles.
2. Linear geometry in the 3-dimensional space: parallel and orthogonal lines, distances, orthogonal projections, angles.
3. Exercises on rotations and translations of the coordinate system.

More theoretical exercises might be asked: e.g., prove or disprove a statement concerning the topics of the course.