GEOMETRY AND ALGEBRA
Learning outcomes of the course unit
Provide students with the instruments for solving linear equation systems, working with matrices, solving simple Analytical Geometry exercises in space, recognising when a matrix is diagonalisable and tracing a quadratic form to a canonical form.
Course contents summary
Operations between sets. Complements. Cartesian product. Relations, functions and their properties.
Vectorial algebra and its applications in geometry.
Vectorial space of real numbers n-tuples. Vectors in a space with n dimensions. Geometric interpretation in a space of dimension n=2 and n=3. Scalar product. Length and norm of a vector. Cauchy-Schwarz inequality. Parallelism and orthogonality between vectors. Projections. Angle between vectors. Fundamental versors. Direction cosines. Linear dependence and linear independence of vectors. Bases. Vector product. Mixed product. Distance between two points. Linear equations. Direction parameters of a line. Equation of a plane. Normal vectors to planes. Angle between planes. Angle between a line and a plane. Distance of a point from a plane. Parallelism and orthogonality between lines and planes. Distance of its skew lines. Distance between planes. Geometric problems in three-dimensional space.
Matrices, determinants, linear systems.
The vector space of matrices. Product (lines by columns) of matrices. Range of matrices. Determinants of square matrices. Determinants and independence of vectors. The formula of the product by the determinants. The co-factor matrix. Matrix inverse and its calculation. The determinant of the matrix inverse of a non-singular matrix. Homogeneous and non-homogenous linear systems. M=n case. Cramer’s rule. Rouchè-Capelli theorem. Gaus-Jordan elimination method. Eigenvalues and eigenvectors. Quadratic forms and their diagonalisation. Applications. Conic and quadratic sections and their canonical forms.
Since this is a basic course, any Linear Algebra or Geometry text covers the subjects dealt with in the lessons.
Procedure for exercises: guided exercises in small groups will be carried out each week.
Examination procedure: during the course two written tests will be given valid for passing the examination, which otherwise entails a written examination and an oral one.