GEOMETRY AND ALGEBRA
cod. 13141

Academic year 2007/08
1° year of course - First semester
Professor
Academic discipline
Geometria (MAT/03)
Field
Formazione scientifica di base
Type of training activity
Basic
60 hours
of face-to-face activities
4 credits
hub:
course unit
in - - -

Learning objectives

Studying sets, linear and vector algebra and its applications in geometry

Prerequisites

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Course unit content

Operations between sets. Complements. Cartesian product. Relations, functions and their properties. Vector algebra and its applications to geometry. Vector space of the n-uples of real numbers. Vectors in a space of n dimensions. geometrical interpretation in spaces of dimension n=2 and n=3. Scalar product. Length and norm of a vector. Caucy-Schwarz inequality. Parallelism and orthogonality between vectors. Projections. Angle between vectors. Basic unit vectors. Direction cosines. Linear dependence and linear independence of vectors. Bases. Vector product. Mixed product. Distance between two points. Straight line equations. Direction parameters of a straight line. Equation of a plane. Plane normal vectors. Angle between planes. Angle between a straight line and a plane. Distance of a point from a plane. Parallelism and orthogonality between straight lines and between planes. Distance between two skew lines. Distance between planes. Geometrical problems in the three-dimensional space. Vector space of matrices. Product (lines per column) of matrices. Rank of matrices. Determinants of square matrices. Determinants and independence of vectors. The product formula for determinants. The cofactor matrix. Inverse matrix and its calculation. The determinant of the inverse matrix of a non-singular matrix. Homogeneous and non-homogeneous linear systems. Case m=n. Cramer's rule. Rouchè-Capelli theorem. Gauss-Jordan elimination method. Eigenvalues and eigenvectors. Quadratic forms and their diagonalization. Applications. Conics and quadrics and their canonical forms. <br />
 

Full programme

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Bibliography

S. Abeasis, Elementi di Algebra lineare e Geometria, Zanichelli

Teaching methods

Theoretical lectures and exercises<br />
written examination

Assessment methods and criteria

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Other information

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