Teacher

prof.ssa aggr. Anna GIORDANO BRUNO

Credits

6 CFU

Language

Italian

Objectives

This course is aimed at providing an introduction to Linear Algebra; in particular, it consists in an elementary treatise about vector spaces, linear maps, matrices, determinants, eigenvalues and eigenvectors, diagonalizability, symmetric and Hermitian matrices, orthogonality, quadratic forms, conics.

Acquired skills

- Knowledge of fundamental algorithms in matrix theory.
- Solution of linear systems.
- Fundamental notions of Linear Algebra.
- Basic notions of Affine and Euclidean Geometry.

Lectures and exercises (topics and specific content)

Preliminaries: sets and maps, equivalence relations; numerical sets; fundamental algebraic structures; the field of complex numbers (4 hours).
Matrices: matrix operations; elementary matrices; invertible matrices and computation of the inverse matrix; rank of a matrix and Kronecker theorem; determinant of a square matrix and Laplace expansion (6 hours).
Linear systems: solution of linear systems, Gauss-Jordan method and Cramer rule; Rouché-Capelli Theorem (8 hours).
Vector spaces: vector spaces and subspaces; bases and dimension; theorems on finite dimensional vector spaces; linear transformations, kernel and image, dimension theorem; Tha matrix associated to a linear transformation; change of basis matrix (16 hours).
Eigenvalues and eigenvectors: computation of eigenvalues, eigenvectors and eigenspaces of a matrix and a linear transformation; diagonalizable matrices and diagonalization of a matrix  (8 hours).
Euclidean spaces: bilinear forms and inner product; Cauchy-Schwarz Theorem; orthogonal and ortonormal bases, Gram-Schmidt procedure; orthogonal complement of a subspace; Spectral Theorem  and its applications (8 hours).
Applications to Geometry: scalar product and vector product; lines and planes; conics, reduction to the canonical form (10 hours).

References

- Edoardo Sernesi, Geometria 1, Bollati Boringhieri
- Domenico Freni, Jung Kyu Canci, Algebra lineare e geometria, Pearson Education
- Keith Nicholson, Algebra lineare, McGraw-Hill Education

Type of exam

Written and oral

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