Physics

SAMOUIL MOLCHO

General aim:
to acquire basic knowledge on linear systems, vector spaces, linear maps, symmetric bilinear forms and hermitian forms.

Specific aim
Knowledge and comprehension: successful students will acquire basic notions and results about solvability of linear systems and its geometric interpretation, matrix calculus, vector spaces and linear maps between them, with special focus on endomorphisms of vector spaces and their associate decompositions into eigenspaces.

Applied knowledge and comprehension: successful students will be able to solve linear systems with a finite number of variables, to
recognize mathematical problems that can be codified into vector spaces and linear maps and therefore to solve them;
to manipulate matrices and determine the solvability of a linear system and the invertibility of a linear map by studying its rank and by
computing the determinant of the associated matrix; moreover, he/she will be able to compute the eigenvalues of a linear endomorphism
and to determine the associated decomposition into eigenspaces.

Critical thinking abilities: in this course the student will acquire basic knowledge that will make him/her able to discover analogies between the topics treated in this course and the topics treated in following courses such as Mechanics and Vectorial Analysis.

Communication skills: ability to illustrate the content of the course in the oral exam and in the possible theoretical questions in the written test.

Learning skills: the learnings of these lectures will allow the student to access the study of the following courses of the degree program in Physics.

The students will be introduced to basic notions and results of linear algebra and geometry.

High school mathematics courses.

Vector spaces.
Subspaces. Sums and direct sums. Fields.
Spanning sets, linearly independent sets, and bases.
Exchange Lemma. Dimension Theorem.
Grassmann’s formula.

Systems of linear equations: coefficient matrix and augmented matrix.
Vector space of solutions of a homogeneous system of linear equations.
Affine space of solutions of a non-homogeneous linear system.
Gaussian elimination. Rank of a matrix.
Rouché–Capelli Theorem. Cartesian and parametric equations of a subspace.

Linear maps: definition and examples. Linear maps \(\mathbb{R}^n \to \mathbb{R}^m\) and \(m \times n\) matrices.
Matrix associated with a linear map.
Image and kernel of a linear map. Rank–Nullity Theorem.
Composition of linear maps and row-by-column matrix multiplication.
Invertible matrices. Computing the inverse of a matrix.
Change-of-basis formulas for the coordinates of a vector and for the matrix of a linear map.

Determinant of a matrix. Binet’s Theorem. Cramer’s formula for the inverse of a matrix.

The diagonalization problem. Definition of eigenvectors and eigenvalues. Eigenvalues as roots of the characteristic polynomial. Algebraic and geometric multiplicity of an eigenvalue. Minimal polynomial. Cayley–Hamilton Theorem.
Introduction to the Jordan Canonical Form Theorem.

Bilinear forms, inner products, and Euclidean inner products: definitions and examples.
Classification of bilinear forms and inner products on an \(n\)-dimensional vector space. Change-of-basis formula for the matrix of a bilinear form.
Sylvester’s Theorem: rank and signature of a bilinear form.
Gram–Schmidt orthogonalization algorithm.
Properties of orthonormal bases.
Orthogonal projections and reflections.
Isometries of Euclidean/Hermitian spaces and orthogonal/unitary matrices.
Self-adjoint endomorphisms and symmetric/Hermitian matrices.
Spectral Theorem: statement and proof.
Criterion for positive-definite matrices.
Quadratic forms and bilinear forms.

Rigid motions and the classification of quadrics in \(\mathbb{R}^n\).
Classification of conic sections in the plane.
Classification of quadric surfaces in space.

Marco Abate, Chiara De Fabritiis. Geometria analitica con elementi di algebra lineare. McGraw-Hill Education
Marco Manetti, Algebra Lineare. Springer

The professor lectures on the board, encouraging active participation from the students.

Not mandatory.

Written and Oral exam.

Show that two bases of a vector space have the same cardinality. Diagonalize a given matrix.