# GEOMETRY

## Learning outcomes of the course unit

BASIC KNOWLEDGE OF LINEAR ALGEBRA AND GEOMETRY.

## Course contents summary

1. Real and complex vector spaces. Linear subspaces: sum and intersection.

Linear combinations of vectors: linear dependence and independence.

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 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. A brief

discussion on the complex case.

7. Three dimensional analytic geometry. Parametric and Cartesian equations

of a line. Mutual position of 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 a plane.

## Course contents

1. Real and complex vector spaces. Linear subspaces: sum and intersection.

Linear combinations of vectors: linear dependence and independence.

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 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. A brief

discussion on the complex case.

7. Three dimensional analytic geometry. Parametric and Cartesian equations

of a line. Mutual position of 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 a plane.

## Recommended readings

ALESSANDRINI, L., NICOLODI, L., GEOMETRIA E ALGEBRA LINEARE, CON ESERCIZI SVOLTI, ED. UNINOVA (PR) 2012.

## Teaching methods

LECTURES.

## Assessment methods and criteria

WRITTEN AND ORAL EXAMINATIONS.