## GEOMETRY AND ALGEBRA

Teachers:
Credits:
9
Site:
PARMA
Year of erogation:
2021/2022
Unit Coordinator:
Disciplinary Sector:
GEOMETRY
Semester:
First semester
Year of study:
1
Language of instruction:

Italian

### Learning outcomes of the course unit

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.

### Course contents summary

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.

### Course contents

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

MATRICES
* 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

COMPLEX NUMBERS
* Cartesian and exponential form of complex numbers
* Operations with complex numbers (sum, product, power, conjugate)
* Norm of a complex number

VECTOR SPACES
* Definition of vector space and subspaces
* Linear combinations
* Linear dependence and independence
* Bases and coordinates
* Sum, direct sum and Grassmann formula

LINEAR MAPS
* Definition
* Kernel and image of a linear map
* Isomorphisms
* Matrices and linear maps

DIAGONALIZATION OF MAPS AND MATRICES
* Eigenvalues and eigenvectors
* The characteristic polynomial
* Diagonalizable matrices
* Algebraic and geometric multiplicities
* Criteria for diagonalizability