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.
Course contents summary
1. Real and complex vector spaces. Linear subspaces: sum andintersection. Linear combinations of vectors: lineardipendence/indipendence. Generators, bases and dimension of a vectorspaces. Grassmann formula for subspaces.
2. Determinants: Laplace expansion and basic properties. Binet theorem.Row and column elementary operations on matrices. Computation of theinverse 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 onlinear maps. Matrix representation of a linear map and change of bases.Isomorphisms and inverse matrix.
5. Endomorphisms of a vector space: eigenvalues, eigenvector andeigenspaces. Characteristic polynomial. Algebraic and geometricmultiplicity. Diagonalizable endomorphisms.
6. Scalar products. Orthogonal complement of a linear subspace.Gram-Schmidt orthogonalization process. Representation of isometries byorthogonal matrices. The orthogonal group. Diagonalization of symmetricmatrices: spectral theorem. Positivity criterion for scalar product:Hurewicz theorem.
7. Two and three dimensional analytic geometry. Parametric andCartesian equations of a line. Mutual position between two lines in thespace; skew lines. Equation of a plane. Canonical scalar product anddistance. Vector product and its fundamental properties. Distance of apoint 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 andaxes.
F. Capocasa, C.Medori: " Algebra Lineare e Geometria Analitica ", Mattioli
A. Nannicini: " Esercizi svolti di algebra lineare, vol.1 ", Pitagora
Examination texts are available at the photocopy-office.