In this course unit students will develop geometrical skills acquired in Linear Algebra I. After a brief summary about eigenvalues, eigenvectors, and linearization, which were already studied in Linear Algebra I, we study the Jordan canonical form. Then, we proceed with the study of inner products. These concepts will be applied to define and compute angles, areas, volumes, to classify conics and quadrics and in metric problems.
At the end, students are expected to be able to deal with concepts, general properties, and computacional agility in relation with the themes of:diagonalization, Jordan canonical forms, and inner products. Students are expected to know how to compute ortonormal bases, diagonalize matrices or transform them into Jordan form, compute angles, areas and volumes, to classify conics and quadrics and to identify lines and planes.
1. Revisions of the concepts of eigenvalues and eigenvectors, and of diagonalization of matrices. Generalized eigenvalues and Jordan canonical forms.
2. Inner products over a real or complex vectorial space: norms, angles and orthogonality; inequality of Cauchy-Schwarz, triangular inequality and theorem of Pitagoras; orthogonal bases, orthonormal bases and the method of Gram-Schmidt; cross product, triple product; areas and volumes; matrix of an inner product.
3. Bilinear applications and quadratic forms: matrix associated to a quadratic form and its classification.
4. Analytic Geometry: affine space; change of referencial; lines, planes, hyperplanes; vectorial equations, parametric equations, cartesian equations; parallelism, orthogonality; metric problems; conics and quadrics.
 CORREIA, Ana Luísa. Álgebra Linear II: support material to be posted in the e-learning platform.