# GEOMETRY

## Learning outcomes of the course unit

The aim of this course is to provide students with essentials tools in Linear Algebra and in Euclidean Geometry in the plane and in the space; students are required also to apply their knowledge and understanding to problems concerning the spatial structure of real environment, graphical and architectonic structures.

The aim of this course is to provide students with essentials tools in Linear Algebra and in Euclidean Geometry in the plane and in the space; students are required also to apply their knowledge and understanding to problems concerning the spatial structure of real environment, graphical and architectonic structures.

## Course contents summary

The course is an introduction to different aspects of Linear Algebra and Geometry. It starts with Euclidean Geometry in the space (vectors, lines, planes), while the second part of the course is devoted to vectors, matrices, linear systems.

In the third part of the course we study vector spaces, linear maps and the diagonalization of linear operators. The course ends with the study of scalar and hermitian products.

The course is an introduction to different aspects of Linear Algebra and Geometry. It starts with Euclidean Geometry in the space (vectors, lines, planes), while the second part of the course is devoted to vectors, matrices, linear systems.

In the third part of the course we study vector spaces, linear maps and the diagonalization of linear operators. The course ends with the study of scalar and hermitian products.

## 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; dimension theorem. 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.

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; dimension theorem. 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

Abate, Marco. Geometria. McGraw-Hill.

Abate, Marco. Geometria. McGraw-Hill.

## Teaching methods

In the lectures we shall propose formal definitions and proofs, with significant examples and applications, and several exercises. Exercises are an essential tool in Linear Algebra and Geometry; they will be proposed also in addition to lectures, in a guided manner, by the “Progetto IDEA”.

In the lectures we shall propose formal definitions and proofs, with significant examples and applications, and several exercises. Exercises are an essential tool in Linear Algebra and Geometry; they will be proposed also in addition to lectures, in a guided manner, by the “Progetto IDEA”.

## Assessment methods and criteria

Learning is checked by: a preliminary multiple-choiche test, a written exam and an oral interview. The student can also perform 2 written exams and tests during the course, to avoid the written exam and test.

In the written exam, through tests and exercises, the student must exhibite basic knowledge related to Geometry and Linear Algebra. In the colloquium, the student must be able to prove properties of the studied structures, using an appropriate geometric and algebraic language and a proper mathematical formalism.

Learning is checked by: a preliminary multiple-choiche test, a written exam and an oral interview. The student can also perform 2 written exams and tests during the course, to avoid the written exam and test.

In the written exam, through tests and exercises, the student must exhibite basic knowledge related to Geometry and Linear Algebra. In the colloquium, the student must be able to prove properties of the studied structures, using an appropriate geometric and algebraic language and a proper mathematical formalism.