B5528 - GEOMETRIA E ALGEBRA LINEARE 2

Learning outcomes

Successful students will acquire a broad understanding of linear algebra notions such as canonical forms of endomorphisms.

They will be able to apply such knowledge to elementary geometric problems.

Course contents

 

Quotient vector spaces. Homomorphism and isomorphism theorem.

Decomposition in generalized eigenspaces. Jordan normal form. Invariants of the Jordan form and Jordan form of companion matrices. Cayley-Hamilton theorem.

Primary decomposition and primary components. Rational normal form. Similarity of matrices is field independent.

Bilinear forms: symmetric, skew-symmetric, (non)degenerate. Real bilinear matrices: positive and negative (semi)definite ones. Signature.

Complex sesquilinear matrices: hermitian, skew-hermitian, positive definite forms. Complex scalar products.

(Right and left) kernel of a bilinear form. Dimension of (right and left) orthocomplement of a subspace. Orthogonal decompositions.

Matrices associated to bilinear forms. Congruence of matrices.

Diagonalization of symmetrich bilinear forms. Decomposition of alternating forms into orthogonal sum of their kernel and symplectic planes.

Complex spectral theorem for normal operators. Real spectral theorem for self-adjoint operatore. Spectral theorem for real symmetric bilinear forms.

Introduction to affine geometry.

 

 

 

 

Readings/Bibliography

M. Manetti "Algebra lineare, per matematici"

Teaching methods

Frontal lectures at the blackboard.

Resolution of exercises in sections.

Students will be given homework that will be corrected and discussed.

Assessment methods

Written and oral exams.

 

Students with specific learning disorders (SLD) or temporary or permanent disabilities are advised to contact the relevant University office well in advance (https://site.unibo.it/studenti-con-disabilita-e-dsa/it). That office will propose any necessary accommodations to the students concerned; however, these proposals must be submitted to the instructor for approval at least 15 days in advance, so that their appropriateness can be assessed in relation to the course's learning objectives.

The use of AI for assessment purposes is prohibited. Any use constitutes a violation of academic integrity.

Teaching tools

Recitation sections.

Office hours

See the website of Nicoletta Cantarini

See the website of Andrea Petracci