# GEOMETRY AND ALGEBRA cod. 1003714

1° year of course - First semester
Professor
Geometria (MAT/03)
Field
"discipline matematiche per l'architettura"
Type of training activity
Basic
40 hours
of face-to-face activities
4 credits
hub: -
course unit
in - - -

Integrated course unit module: MATHEMATICS

## Learning objectives

Knowledge and understanding:
basic theory of vector spaces and geometry of space.

Applying knowledge and understanding:
a) solve systems of linear equations;
b) diagonalize (symmetric) matrices;
c) solve easy problems of analytic geometry;
d) recognize the type of a conic and write its canonical form.

Making judgements: basic.

Communication and learning skills:
properly express themselves with mathematical language.

## Prerequisites

Precourse of Mathematics
(if possible).

## Course unit content

1. Real and complex vector spaces.

2. Determinants and rank of a matrix.

3. Linear systems.

4. Linear applications.

5. Endomorphisms of a vector space.

6. Scalar products.

7. Affine geometry of space.

8. Elements of analytic geometry of the three-dimesional space.

## Full programme

1. Real 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.
7. Affine and projective geometry.
8. Two and 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.
9. Complements of geometry.

## Bibliography

F.Capocasa, C.Medori: "Corso di Geometria e Algebra Lineare", ed. S.Croce (Parma, 2013).

## Teaching methods

Lessons (on the blackboard).

## Assessment methods and criteria

Written examination (preceded by a test)
and oral examination (on demand).

## Other information

Lecture attendance is compulsory.