## Vector and Euclidean Spaces

### Teaching unit

### Description

### Objectives

To know the fundamentals and applications of the vector spaces

To know the Euclidean Geometry and to be able to carry out the corresponding calculations

To acquire and to apply, of autonomous form and to interdiscipline, new concepts and methods related to the subject and to use adequately the usual terminology of the subject.

To use diverse technological tools (as IT software) that facilitate the resolution of mathematical problems and to understand the limitations of the above mentioned tools.

To know what is a mathematical proof and his differences with other types of reasonings. To understand and to apply correctly different types of mathematical arguments: beginning of mathematical induction, reduction to the absurdity, reciprocal theorem, etc.

### Subjects

#### Lecture topics

- Vector space structure. Subspaces
- Linear combinations of vectors
- Vectors linear independient
- Basis and dimension of a vector space
- Scalar products
- Norm of a vector
- Orthogonal and Orthonormal Bases
- orthogonal complement
- Orthogonal projection

#### Laboratory topics

- Subspaces in Rn