GEOMETRY AND ALGEBRA
Learning outcomes of the course unit
Knowledge and understanding:
basic Linear Algebra and Geometry of the space.
Applying knowledge and understanding:
a) solve systems of linear equations;
b) diagonalize matrices;
c) solve easy problems of analytic geometry;
d) recognize the type of a conic and write its canonical form.
Communication and learning skills:
properly express themselves with mathematical language.
Course contents summary
This course is an introduction to different aspects of Algebra, Linear Algebra and Geometry.
The first part is devoted to Euclidean Geometry in the space (vectors, lines, planes), while the second part of the course is devoted to matrices and linear systems. In the third part we study vector subspaces of Rn, linear maps and the diagonalization of linear operators.
Euclidean Geometry in the space.
1. Vectors and its operations. Coordinates. Scalar product. Distances and angles. Vector product in R3.
2. 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. Quadric surfaces.
Vectors, matrices, linear systems.
3. The n-dimensional space Rn and its properties.
4. Matrices and their properties. Determinants: Laplace expansion and basic properties. Binet theorem. Row and column elementary operations on matrices. Computation of the inverse matrix. Rank of a matrix.
5. Linear systems: Gauss-Jordan method and Rouché Capelli theorem.
6. Linear subspaces of Rn. Linear combinations of vectors: linear dipendence/indipendence. Generators, bases and dimension of a vector subspace.
7. Linear maps. Definition of kernel and image. Matrix representation of a linear map. Isomorphisms and inverse matrix.
8. Eigenvalues, eigenvector and eigenspaces. Characteristic polynomial. Algebraic and geometric multiplicity. Diagonalizable operators.
ALESSANDRINI, L., NICOLODI, L., GEOMETRIA A, ED. UNINOVA (PR) 2004.
In the lectures we shall propose formal definitions and proofs, with significant examples and applications, and several exercises. Exercises are an essential tool in Linear Algebra; they will be proposed also in addition to lectures, in a guided manner.
Assessment methods and criteria
Learning is checked by a written exam. The student can also perform two written exams during the course, to avoid the final written exam.
In the written exam the student must exhibit basic knowledge related to Linear Algebra and Euclidean Geometry in the space.