Università Roma Tre

Vector spaces over arbitrary fields
matrices. Sum and product of matrices.
Vector space of matrices.
Symmetric, antisymmetric, diagonal and triangular matrices.
Elementary matrices .
Invertibili matrices. Gauss Jordan method for the inverse.
Linear systems .
The vector space of solutions of a homogeneous system.
Gauss algorithm.

Lineari dependence.
Bases for vector space.
Dimension.
Linear maps
Matrices for linear maps.
Grassmann Formula .
Kernel and image.
Dimension theorem.
Rank of a matrix.
Rouché-Capelli theorem.

Determinant of a matrix: geometric meaning.
Recursive definition of the Determinant (Laplace).
Rank and Determinant.
Determinant properties and axiomatic definition.
Determinant of the product.
Permutation matrices and their sign.
Explict formula for the determinant.

Change of basis matrix in a vector space.
Change of coordinate matrix in a vector space.
The set of bases of a vector space.
The matrix of a linear application with respect to two bases.
Change of basis and linear applications.

Linear operators.
Similar matrices.
Invariant subspaces for linear operators.
Eigenvectors and eigenvalues of linear operators.
Characteristic polynomial.
Diagonalization and triangularization.
Algebraic and geometric multiplicity of eigenvectors and eigenvalues.
Duale vector spac, dual basis.

Geometry in affine spaces.
Affine subspaces: lines, planes, hyperplanes.
Cartesian and parametric equations.
Parallelism.

Edoardo Sernesi - Geometria 1 - Bollati Boringhieri

Michael Artin - Algebra - Bollati Boringhieri