Knowledge and understanding:
the theory of vector spaces.
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:
evaluate the correctness of a simple proof.
Communication and learning skills:
properly express themselves with mathematical language.
Precourse. This exam is preparatory to "Analisi matematica 2".
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.
9. Complements of algebra/geometry.
0. Preliminaries: equivalence relations and partitions; algebraic structures
(groups and fields).
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: Sylvester theorem.
Outline of the complex case.
7. Affine geometry.
Parallelism and mutual position of affine subspaces.
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 algebra/geometry.
F. Capocasa, C.Medori: "Corso di Geometria", ed. S.Croce (Parma, 2013).
Lessons.
Written examination (preceded by a test) and oral examination.
The examamination has to ensure the intellectual maturity of the candidate and his organic preparation on the arguments of the course.
Lecture attendance is highly recommended.
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