Physics

GABRIELE MONDELLO
FEDERICO PELLARIN

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.

At the end of the course the student will know basic linear algebra and basic quadratic forms, and will be able to solve elementary problems in linear algebra and in quadratic forms.
Thanks to the assignment of weekly sheets of exercises, the student will be able to understand given proofs and to determine their rigour, and will be to able to produce simple proofs in linear algebra and to illustrate them with method and clarity, also replying to possible objections.
Moreover the student will achieve the ability to provide solutions to new problems and to evaluate such solutions with logic rigour.

No prerequisite is needed.

Basic notions. Sets, functions, fields, polynomials. The field of complex numbers.
Linear systems. Gaussian elimination, structure of solutions.
Vector spaces. Linear combinations, subspaces and affine subspaces, linear independence, bases and generators, dimension, sum and intersection, Grassmann's formula.
Linear maps. Kernel, image, rank–nullity theorem.
Matrices. Algebra of matrices, matrix associated to a linear map, rank, invertible matrices, change of coordinates, determinants.
Diagonalization. Eigenvalues and eigenvectors, characteristic polynomial.
Bilinear forms. Scalar products, Hermitian products, orthogonal bases, Sylvester's law of inertia, Euclidean and Hermitian vector spaces.
Self-adjoint operators, linear isometries, Spectral Theorem.

Marco Abate e Chiara de Fabritiis, Geometria Analitica con elementi di Algebra Lineare, III edizione (2015), ed. Mc Graw Hill Educational

In the written test the student will be asked to solve some exercises.
In the oral exam the student will be asked to discuss problems and to illustrates how the theory may help in solving such problems.

Optional but strongly encouraged.

To pass the exam, a grade of not less than 18/30 must be obtained. The student has to prove to have achieved a sufficient knowledge of the topics treated in class and has be able to solve at least the simplest exercises among the assigned ones.
To achieve a score of 30/30 cum laude, the student has to prove to have achieved an excellent knowledge of all the topics treated in class and has to be able to link them in a logical and consistent way.

Find all complex numbers that solve the following equation.
Determine whether the following subset is a vector subspace.
Determine a basis of the following vector (sub)space.
Determine the matrix that represents the following linear map with respect to given bases of the domain and of the codomain.
Determine whether the following endomorphism is triangularizable or diagonalizable.
Compute a basis that diagonalizes the following quadratic form.
Compute the adjoint of the following homomorphism.
Find an orthonormal basis of the following self-adjoint linear transformation.

• Fields, complex numbers, polynomials


• Vector spaces and subspaces, intersection


• Sum of vector subspaces, linear combinations, span


• Linear independence, bases, dimension


• Direct sum of subspaces, Grassmann formula


• Linear maps, kernel and image, composition and inverse


• Linear systems of equazions, Gauss elimination, Rouché-Capelli theorem


• Linear coordinates, algebra of matrices, Hom(V,W)


• Dual vector space, bases and dimension, zero locus and annihilator


• Determinant, characterization, Binet theorem, Cramer theorem and inverse matrix


• Endomorphisms, characteristic polynomial, eigenvalues and eigenspaces, algebraic and geometric multiplicities


• Triangulability criterion for endomorphisms


• Diagonalizability criteria for endomorphisms


• Symmetric bilinear forms, radicale, matrix representation, positive definite bilinear forms (norm, distance, Cauchy-Schwarz and triangular inequality, angles)


• Linear map from V to its dual associated to a bilinear form and its rank, orthogonal to a subset, nondegenerate bilinear forms and duality


• Diagonalizability of symmetric bilinear forms, classification of symmetric bilinear forms in the complex case (rank) and in the real case (rank and signature)


• Adjoint (wrt to assigned symmetric bilinear forms) of a linear map, self-adjoint endomorphisms, spectral theorem


• Real symmetric bilinear forms, criteria for computing their indices


• Extension of the above result to the case of Hermitian forms


• Classification of orthogonal and of unitary endomorphisms