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Geometry ( 9 CFU ) Finalità

Supply the student with tools for:

a) solve systems of linear equations;

b) diagonalize (symmetric) matrices;

c) solve easy problems of analytic geometry;

d) Operations on vectors and matrices.

a) solve systems of linear equations;

b) diagonalize (symmetric) matrices;

c) solve easy problems of analytic geometry;

d) Operations on vectors and matrices.

Programma

1. Real and complex vector spaces. Linear subspaces: sum and intersection. Linear combinations of vectors: linear dipendence/indipendence. Generators, bases and dimension of a vector spaces. Grassmann formula for subspaces.

2. Determinants: Laplace expansion and basic properties. Binet theorem. Row and column elementary operations on matrices. Computation of the inverse matrix. Rank of a matrix.

3. Linear systems: Gauss-Jordan method and Rouché Capelli theorem.

4. Linear maps. Definition of kernel and image; fundamental theorem on linear maps. Matrix representation of a linear map and change of bases. Isomorphisms and inverse matrix.

5. Endomorphisms of a vector space: eigenvalues, eigenvector and eigenspaces. Characteristic polynomial. Algebraic and geometric multiplicity. Diagonalizable endomorphisms.

6. Scalar products. Orthogonal complement of a linear subspace. Gram-Schmidt orthogonalization process. Representation of isometries by orthogonal matrices. The orthogonal group. Diagonalization of symmetric matrices: spectral theorem. Positivity criterion for scalar product. A brief discussion on the complex case..

7. Three dimensional analytic geometry. Parametric and Cartesian equations of a line. Mutual position between two lines in the space; skew lines. Equation of a plane. Canonical scalar product and distance. Vector product and its fundamental properties. Distance of a point from a line and from a plane.

8. Topics in algebra and/or geometry.

2. Determinants: Laplace expansion and basic properties. Binet theorem. Row and column elementary operations on matrices. Computation of the inverse matrix. Rank of a matrix.

3. Linear systems: Gauss-Jordan method and Rouché Capelli theorem.

4. Linear maps. Definition of kernel and image; fundamental theorem on linear maps. Matrix representation of a linear map and change of bases. Isomorphisms and inverse matrix.

5. Endomorphisms of a vector space: eigenvalues, eigenvector and eigenspaces. Characteristic polynomial. Algebraic and geometric multiplicity. Diagonalizable endomorphisms.

6. Scalar products. Orthogonal complement of a linear subspace. Gram-Schmidt orthogonalization process. Representation of isometries by orthogonal matrices. The orthogonal group. Diagonalization of symmetric matrices: spectral theorem. Positivity criterion for scalar product. A brief discussion on the complex case..

7. Three dimensional analytic geometry. Parametric and Cartesian equations of a line. Mutual position between two lines in the space; skew lines. Equation of a plane. Canonical scalar product and distance. Vector product and its fundamental properties. Distance of a point from a line and from a plane.

8. Topics in algebra and/or geometry.

Modalità d'esame

Usually the evaluation consists of a written exam.

Propedeuticità

Precourse.

Testi consigliati

F. Capocasa, C.Medori: Corso di Geometria, ed. S. Croce.

A. Alessandrini, L.Nicolodi: Geometria A.

A. Alessandrini: Geometria B.

A. Alessandrini, L.Nicolodi: Geometria A.

A. Alessandrini: Geometria B.