# LINEAR ALGEBRA

# LINEAR ALGEBRA

Basic algebra operazions, logarithms, exponential functions.

Basics on Trigonometry. Basics on Euclidean geometry. Basics on Logics.

Ability to manipulate algebraic calculus and to develop the skill to apply logic reasonings to several circumstances.

Elements of Logics. Basics on set theory. Notions of fields, rings, commutative groups. The field of complex numbers. Algebraic equations in one unknown. The Fundamental Theorem of Algebra.

Vector spaces. The space R^n. Linear subspaces and affine subspaces.

Cartesian and Parametric representations of linear and affine subspaces.

Linear systems: resolution with Gauss-Jordan method. Dependence and linear independence. Basis and dimension of linear subspaces. Rank of a matrix. Change of basis. Inverse of a matrix. Direct sum and linear complements of subspaces.

Affine Geometry ( parallelism, generating sets, reference systems). Inner products and norms. Orthogonal bases. Orthogonal complements.

Euclidean Geometry (in R^n): normal vector of an hyperplane; orthogonality relation among subspaces; distance between a point and an affine subspace.

Linear tranformations. Representative matrix of a given linear transformation. Kernel of a linear transformation. Invariant subspaces. Eigenspaces, eigenvectors, eigenvalues. Hermitian and anti-hermitian transformations. Orthogonal and unitary transformations. Determinants. Characteristic polynomial. Jordan Canonical form. Affine transformations.

Diagonalization over the real numbers and over the complex numbers.

Bilinear forms. Symmetric bilinear forms. Quadratic forms. Conics and quadrics.

Direct and iterative methods for the resolution of linear systems. Power iteration to find eigenvectors. Basics on the numerical resolutions of algebraic equations.

Elements of Logics. Basics on set theory. Notions of fields, rings, commutative groups. The field of complex numbers. Algebraic equations in one unknown. The Fundamental Theorem of Algebra.

Vector spaces. The space R^n. Linear subspaces and affine subspaces.

Cartesian and Parametric representations of linear and affine subspaces.

Linear systems: resolution with Gauss-Jordan method. Dependence and linear independence. Basis and dimension of linear subspaces. Rank of a matrix. Change of basis. Inverse of a matrix. Direct sum and linear complements of subspaces.

Affine Geometry ( parallelism, generating sets, reference systems). Inner products and norms. Orthogonal bases. Orthogonal complements.

Euclidean Geometry (in R^n): normal vector of an hyperplane; orthogonality relation among subspaces; distance between a point and an affine subspace.

Linear tranformations. Representative matrix of a given linear transformation. Kernel of a linear transformation. Invariant subspaces. Eigenspaces, eigenvectors, eigenvalues. Hermitian and anti-hermitian transformations. Orthogonal and unitary transformations. Determinants. Characteristic polynomial. Jordan Canonical form. Affine transformations.

Diagonalization over the real numbers and over the complex numbers.

Bilinear forms. Symmetric bilinear forms. Quadratic forms. Conics and quadrics.

Direct and iterative methods for the resolution of linear systems. Power iteration to find eigenvectors. Basics on the numerical resolutions of algebraic equations.

Elementi di Algebra e Geometria Vol.1, A. Pasini, Liguori Editore, 1998.

Elementi di Algebra e Geometria Vol.2, A. Pasini, Liguori Editore, 1998.

Elementi di Algebra e Geometria Vol.3, A. Pasini, Liguori Editore, 1998.

A. Quarteroni, R. Sacco, F. Saleri Matematica Numerica, Springer, 2008.

Lectures.Exercises.

Written and oral tests.

On the web page of the course one can download some files with some topics on Numerical calculus.