29228 - Geometry and Algebra T

Learning outcomes

Provide the main tools of linear algebra ( in particular matrices , vector spaces , linear systems , eigenvalues , quadratic forms ) and their application in the geometric environment, ensuring both the understanding of the links between the different parts of the theory , and the operational capability.

Readings/Bibliography

Real numbers and complex numbers
--- Operations on complex numbers
--- Roots of complex numbers.

Linear algebra.
- Matrices and linear systems
--- Gauss method for the solution of a linear system
--- Rank of a matrix and solvability of a linear system
- Vector spaces and linear maps
--- Subspaces, bases of vector spaces
--- Dimension of a vector space
--- Linear maps; kernel and image of a linear map.
--- The matrix associated to a linear map.
- Endomorphisms
--- Eigenvalues and eigenvectors
--- Characteristic polynomial
--- invertible linear maps and matrices
--- Diagonalizable endomorphisms
- Bilinear forms and inner products
--- Matrix associated to a bilinear form
--- Inner products and Euclidean spaces

Analytic geometry of the Euclidean plane and the ordinary Euclidean space
---- Extensions of the ordinary space
- Lines in the plane and theis equations.
--- Incidence conditions and angle between two lines - Conics
--- Conics and associated quadratic forms
--- Tangent lines to a conic
--- Geometric elements of a conic
- Lines and plans in the ordinary space
- Elements on quadrics and associated quadratic forms

 

Teaching methods

Frontal lessons

Assessment methods

Written and oral exam

Office hours

See the website of Sergio Venturini