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) recognize the type of a conic and write its canonical form.
Precourse. This exam is preparatory to "Analisi matematica 2".
Course contents summary
Linear algebra and analytic geometry.
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: Hurewicz theorem.
7. 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.
8. Conics: elementary properties. Affine and Euclidean classifications. Affine invariants and canonical form of a conic. Center of symmetry and axes.
F. Capocasa, C.Medori: "Corso di Geometria", ed. S.Croce.
Lessons (on the blackboard).
Assessment methods and criteria
Written examination (preceded by a test)
and oral examination.