GEOMETRY AND ALGEBRA
The course not only aims to give to the student the basic tools of linear algebra and geometry, but also aims to transmit to the student the methods and the language of mathematics, which can be used also in other fields.
The course is an introduction to Linear Algebra. In particular, it will cover the following topics: vectors and vector spaces, matrices, linear systems and linear maps; analytic gempetry in the three-dimensional space, lines, planes and their mutual positions.
VECTORS IN THREE-DIMENSIONAL SPACE
* Vectors, coordinates and componentwise operations
* Scalar product, length, distance and orthogonality, the Cauchy--Schwartz inequality and angles
* Vector product
PLANES AND LINES
* Planes, lines and their equations
* Mutual positions
THE N-DIMENSIONAL SPACE
* Operations with vectors, scalar product and orthogonality, length and distance, angle between two vectors
* Definition and operations with matrices (sum and product)
* Invertible matrices
* Transpose of a matrix
* Determinant and rank of a matrix
LINEAR SYSTEMS AND MATRICES
* Elementary operations
* Solutions of a system
* Gauss algorithm
* Cramer rule
* Rank and solutions of a linear system
* Cartesian and exponential form of complex numbers
* Operations with complex numbers (sum, product, power, conjugate)
* Norm of a complex number
* Definition of vector space and subspaces
* Linear combinations
* Linear dependence and independence
* Bases and coordinates
* Sum, direct sum and Grassmann formula
* Kernel and image of a linear map
* Matrices and linear maps
DIAGONALIZATION OF MAPS AND MATRICES
* Eigenvalues and eigenvectors
* The characteristic polynomial
* Diagonalizable matrices
* Algebraic and geometric multiplicities
* Criteria for diagonalizability
* L. Alessandrini, L. Nicolodi; "GEOMETRIA A"; Uni.Nova (Parma, 2002).
* L. Alessandrini, L. Nicolodi; "GEOMETRIA E ALGEBRA LINEARE, con esercizi svolti"; Uni.Nova (Parma 2012).
Mainly lessons in classroom. Together with the theory, some significative examples will be diecussed, and there will be also some exercises.
The examination consists in a written exam and in an ora lest. During the course there will be two intermediate tests: passing both of them is equivalent to have passed the written exam.