GEOMETRY AND ALGEBRA
Learning outcomes of the course unit
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.
Precourse of Mathematics
Course contents summary
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.
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.
F.Capocasa, C.Medori: "Corso di Geometria e Algebra Lineare", ed. S.Croce (Parma, 2013).
Lessons (on the blackboard).
Assessment methods and criteria
Written examination (preceded by a test)
and oral examination (on demand).
Lecture attendance is compulsory.