Acquire a good knowledge of the concepts and methods of basic linear algebra, with particular attention given to the study of linear systems, matrices and determinants, vector spaces and linear applications, affine geometry.
Curriculum
teacher profile teaching materials
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.
Michael Artin - Algebra - Bollati Boringhieri
Programme
Vector spaces over arbitrary fieldsmatrices. 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.
Core Documentation
Edoardo Sernesi - Geometria 1 - Bollati BoringhieriMichael Artin - Algebra - Bollati Boringhieri
Reference Bibliography
Edoardo Sernesi - Geometria 1 - Bollati Boringhieri Michael Artin - Algebra - Bollati BoringhieriType of delivery of the course
Lectures are taught live in class and are also available on line. Lectures are filmed and students are allowed to record themType of evaluation
Written exams are made of exercises. Oral exams are about theory and examples. teacher profile teaching materials
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.
Michael Artin - Algebra - Bollati Boringhieri
Programme
Vector spaces over arbitrary fieldsmatrices. 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.
Core Documentation
Edoardo Sernesi - Geometria 1 - Bollati BoringhieriMichael Artin - Algebra - Bollati Boringhieri
Reference Bibliography
Edoardo Sernesi - Geometria 1 - Bollati Boringhieri Michael Artin - Algebra - Bollati BoringhieriType of delivery of the course
Lectures are taught live in class and are also available on line. Lectures are filmed and students are allowed to record themType of evaluation
Written exams are made of exercises. Oral exams are about theory and examples.