Linear Algebra
I) Groups, rings, fields. Z, K[x], C.
II) Matrices over a field. Matrices addition, scalar multiplication, abelian group of matrices, matrix multiplication (or product). Properties. Ring of square matrices. Diagonal, triangular, scalar , symmetric, skew-simmetric matrices and transpose of matrix.
III) Vector spaces and their properties over a filed K. Examples: K[x], Kn, Km,n.. Subspaces. Intersection and sum of vector spaces. Direct sum. Linear combinations. Span, Linear Independence and dependence,Finitely generated vector spaces, Base, Dimension. Steinitz’s Lemma *, Grassmann’s formulas*.
IV) Determinants and their properties. Theorems of Binet*,Laplace I*, Laplace II*, Adjunct matrix, Inverse, Rank and Reduction of a matrix. Theorem of Kronecker*. Systems of linear equations. Rouchè-Capelli‘s rule, Cramer’s rule. Solving systems of linear equations.
V) Linear maps between vector spaces and their properties. Kernel and image of a linear map. Injective, surjective maps and isomorphisms. Study of linear maps. Matrices associated to linear maps. Change of base matrix. Similar matrices.
VI) Eigenvalues, Eigenvectors and Eigenspaces of a matrix. Characteristic polynomial. Dimension of an eigenspace. Relation between Algebraic multiplicity and geometric multiplicity. Linear Independence of the eigenvectors. Diagonalizable linear maps and diagonalization of a matrix.
Geometry
I)Euclidean (geometric) vectors and their properties. Scalar multiplication, dot (or scalar) product, wedge (or cross) product.
II)Cartesian coordinates. Points, lines , Homogeneous coordinates, Points at infinity (Improper Points), Parallel and orthogonal Lines. Slope of a line. Distances from a point to a line. Pencil of lines. Planes in The space. Coplanar and Skew lines. Pencil of Planes. Angles between lines and planes. Distance from a point to a plane and from a point to a line in the space.
III) Conics and their associated matrices. Orthogonal Invariants. Canonical reduction of a conic*. Irreducible and degenerate conics. Rank of its associated matrix. Discriminant of a conic. Parabolas, Ellipses, Hyperbolas: equations, focus, eccentricity, directrix, semi-maior axis, center. Circumferences, Tangents, and pencils of conics.
IV) Quadrics and its associated matrix. Nondegenerate, degenerate and singular quadric surfaces. Cones and cylinders. Classification. Rulings on a quadric, plane sections of a quadric.
1) S. Giuffrida, A.Ragusa, Corso di Algebra Lineare, Ed. Il Cigno G.Galilei, Roma 1998 (Linear Algebra).
2) G. Paxia, Lezioni di Geometria, Spazio Libri, Catania, 2005 (Geometria) available at www.giuseppepaxia.com