# GEOMETRY

## Learning outcomes of the course unit

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.

## Course contents summary

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-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 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. 8. Topics in algebra and/or geometry.

## Course contents

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-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 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. 8. Topics in algebra and/or geometry.

## Recommended readings

L. Alessandrini, L. Nicolodi,

Geomeria e Algebra Lineare con esercizi svolti,

Editrice UNI.NOVA, Parma, 2012.

S. Lang, Linear Algebra, 3rd ed.,

Undergraduate Texts in Mathematics, Springer-Verlag, 2004.

## Teaching methods

Class lectures and recitation sessions

## Assessment methods and criteria

Written and oral exam