GEOMETRY AND ALGEBRA
Learning outcomes of the course unit
Studying sets, linear and vector algebra and its applications in geometry
Course contents summary
Operations between sets. Complements. Cartesian product. Relations, functions and their properties. Vector algebra and its applications to geometry. Vector space of the n-uples of real numbers. Vectors in a space of n dimensions. geometrical interpretation in spaces of dimension n=2 and n=3. Scalar product. Length and norm of a vector. Caucy-Schwarz inequality. Parallelism and orthogonality between vectors. Projections. Angle between vectors. Basic unit vectors. Direction cosines. Linear dependence and linear independence of vectors. Bases. Vector product. Mixed product. Distance between two points. Straight line equations. Direction parameters of a straight line. Equation of a plane. Plane normal vectors. Angle between planes. Angle between a straight line and a plane. Distance of a point from a plane. Parallelism and orthogonality between straight lines and between planes. Distance between two skew lines. Distance between planes. Geometrical problems in the three-dimensional space. Vector space of matrices. Product (lines per column) of matrices. Rank of matrices. Determinants of square matrices. Determinants and independence of vectors. The product formula for determinants. The cofactor matrix. Inverse matrix and its calculation. The determinant of the inverse matrix of a non-singular matrix. Homogeneous and non-homogeneous linear systems. Case m=n. Cramer's rule. Rouchè-Capelli theorem. Gauss-Jordan elimination method. Eigenvalues and eigenvectors. Quadratic forms and their diagonalization. Applications. Conics and quadrics and their canonical forms.
S. Abeasis, Elementi di Algebra lineare e Geometria, Zanichelli
Theoretical lectures and exercises