# GEOMETRY cod. 13102

1° year of course - Second semester
Professor
Geometria (MAT/03)
Field
Matematica, informatica e statistica
Type of training activity
Basic
72 hours
of face-to-face activities
9 credits
hub:
course unit
in - - -

## Learning objectives

BASIC KNOWLEDGE OF LINEAR ALGEBRA AND GEOMETRY.

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## Course unit content

1. Real and complex vector spaces. Linear subspaces: sum and intersection.
Linear combinations of vectors: linear dependence and independence.
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 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 of 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 a plane.

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## Bibliography

ALESSANDRINI, L., NICOLODI, L., GEOMETRIA E ALGEBRA LINEARE, CON ESERCIZI SVOLTI, ED. UNINOVA (PR) 2012.

LECTURES.

## Assessment methods and criteria

WRITTEN AND ORAL EXAMINATIONS.

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