# GEOMETRY

## Learning outcomes of the course unit

Supply the student with tools for:

a) solve systems of linear equations;

b) diagonalize (symmetric) matrices;

c) solve easy problems of analytic geometry;

d) recognize the type of a conic and write its canonical form.

## Prerequisites

Precourse. This exam is preparatory to "Analisi matematica 2".

## Course contents summary

Linear algebra and analytic geometry.

## Course contents

1. Real and complex vector spaces. Linear subspaces: sum and intersection. Linear combinations of vectors: linear dipendence/indipendence. Generators, bases and dimension of a vector spaces. Grassmann formula for subspaces.

2. Determinants: Laplace expansion and basic properties. Binet theorem. Row and column elementary operations on matrices. Computation of the inverse matrix. Rank of a matrix.

3. Linear systems: Gauss-Jordan method and Rouché Capelli theorem.

4. Linear maps. Definition of kernel and image; fundamental theorem on linear maps. Matrix representation of a linear map and change of bases. Isomorphisms and inverse matrix.

5. Endomorphisms of a vector space: eigenvalues, eigenvector and eigenspaces. Characteristic polynomial. Algebraic and geometric multiplicity. Diagonalizable endomorphisms.

6. Scalar products. Orthogonal complement of a linear subspace. Gram-Schmidt orthogonalization process. Representation of isometries by orthogonal matrices. The orthogonal group. Diagonalization of symmetric matrices: spectral theorem. Positivity criterion for scalar product: Hurewicz theorem.

7. Two and three dimensional analytic geometry. Parametric and Cartesian equations of a line. Mutual position between two lines in the space; skew lines. Equation of a plane. Canonical scalar product and distance. Vector product and its fundamental properties. Distance of a point from a line and from a plane.

8. Conics: elementary properties. Affine and Euclidean classifications. Affine invariants and canonical form of a conic. Center of symmetry and axes.

## Recommended readings

F. Capocasa, C.Medori: "Corso di Geometria", ed. S.Croce.

## Teaching methods

Lessons (on the blackboard).

## Assessment methods and criteria

Written examination (preceded by a test)

and oral examination.