Learning objectives
The goal of the course is an introduction of the linear algebra with a particular interest in the solution of a system of linear equations.
Prerequisites
this course is self-contained
Course unit content
geometry and linear algebra
Full programme
Elements of analytic geometry of the 3-dimensional space. Parametric and cartesian equations
Parametric and Cartesian of a straight line. Mutual position of two
lines. Equation of a plane. Scalar product and distance. Wedge product and its fundamental properties.
Real and complex vector spaces. Subspaces: sum and intersection. Linear combination of vectors:
linear dependence / independence. Generators, bases and dimension of
a vector space. Grassmann formula.
Determinants: definition using the formulas of Laplace and
fundamental properties. Binet theorem. Elementary operations
of the row and column of a matrice. Calculation of the inverse matrix.
Rank of
a matrix.
System of linear equations: Gauss-Jordan's theorem and Theorem of Rouche-Capelli
Linear applications. Definition of the kernel and of image, Dimension's theorem, matrix associated to a linear application and rule base change.
Isomorphisms.
Endomorphisms of a vector space: eigenvalues, eigenvectors and
eigenspaces. Characteristic polynomial. Algebraic multiplicity and
geometry of an eigenvalue. Diagonalizable endomorphisms.
Scalar products. Orthogonal complement of a subspace.
Process of Gram-Schmidt orthogonalization. The orthogonal group.
Diagonalization of symmetric matrices: the spectral theorem.
Positivity criterion for scalar products. Outline of the complex case.
Bibliography
abate-De Fabritiis
"Geometria e algebra lineare"
Teaching methods
class
Assessment methods and criteria
written and oral exam
Other information
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