# 81855 - GEOMETRIA 1A

## Learning outcomes

At the end of the course students know the basic concepts of linear algebra (matrices, linear systems, vector spaces, linear applications), and how to apply them to the solution of some problems of analytic geometry.

## Course contents

Vector space over a field. Examples of fields. The rational field, the real field, the complex field, finite fields. Examples of vector spaces: the numeric space K^n, the space of matrices, the space of polynomials with coefficients in a field. The space of functions on a set with values in a field. The space of geometric vectors in dimension 1,2,3. Some elementary consequences of the axioms of vector spaces. Vector subspaces: intersection of vector subspaces, sum of vector subspaces. Cartesian product of vector spaces. Quotient of a vector space with respect to a subspace. Linear combinations of vectors. Subspace generated by a subset of a vector space. Linear dependence and independence. Generators of a vector space. Finitely generated vector spaces. Basis of a vector space. Coordinates of a vector with respect to a base. Existence of a base in a vector space. Base completion theorem. All the bases of a vector space have the same cardinality. Grassmann formula. Direct sum. Dimension of the quotient space. Linear applications. Examples of linear applications. Linear application between numeric spaces associated to a matrix. Special case: the dual space. The set of linear applications is in a natural way a vector space. The composition of linear applications is linear. Gauss reduction. Matrix product. Elementary operations as products with particular matrices. Isomorphisms of vector spaces. Construction of some isomorphisms of vector spaces. Two spaces (on the same field) with the same dimension are isomorphic. A linear application is determined by the values it takes on a basis. Dimension of the space of  linear applications. Kernel and image of a linear application. Isomorphism V / Ker = Im. Preimage of a vector by a linear application. Matrix associated to a linear application with respect to a choice of bases. Its properties. Dependence on the choice of the bases. Matrix of the composition of linear applications. Determination of the kernel and image of linear applications associated with matrices. Invertible matrices. Applications of the theory carried out so far to linear systems. Rank of a matrix. Rouche'-Capelli theorem. Determinant of a square matrix. Characterization, calculation, behaviour with respect to elementary operations on rows and columns. Binet theorem. Laplace development of the determinant. Characterization of invertible matrices. Cramer formula for the resolution of square systems. Applications of linear algebra to analytic geometry: lines and planes in space, their mutual position. Vector calculus in three-dimensional space. The vector product.

## Readings/Bibliography

M. Manetti. Algebra Lineare per Matematici.

http://www1.mat.uniroma1.it/people/manetti/AL2017/algebralineare2017.pdf

S. Lang. Algebra Lineare. Bollati Boringhieri

## Teaching methods

Teaching at the blackboard.

## Assessment methods

Exam consisting of a written test and an oral test.

Students are admitted to the oral exam only if the written exam is sufficient. Both tests must be held in the same exam session.

The written exam consists of a section with multiple-choice questions and 3 exercises to be solved. The exercises will be corrected  only if all  the answers to the multiple choice questions are correct.

The answers to the multiple-choice questions and the solution of the exercises will be available on the teachers' web page.

## Office hours

See the website of Luca Migliorini

See the website of Nicoletta Cantarini