Exam results

WINTER SESSIONS

Results of written exam 18/02/19

Text of written exam 18/02/19

Results of written exam 04/02/19

Text of written exam 04/02/19

SEPTEMBER SESSIONS

Results of written exam 18/09/18

Text of written exam 18/09/18

Results of written exam 06/09/18

Text of written exam 06/09/18

SUMMER SESSIONS

Results of written exam 02/07/18

Text of written exam 02/07/18

Results of written exam 18/06/18

Text of written exam 18/06/18

One is admitted to the oral exam only if the written exam is sufficient (at least 18/30).

Exam dates

WINTER SESSIONS

First Session

Written exam: Monday 04/02/19, 15.00, Aula C3

Oral exam: Thursday 07/02/19, 10.00, Aula C5

Second Session

Written exam: Monday 18/02/19, 15.00, Aula C3

Oral test: Thursday 21/02/19, 10.00, Aula C5

SEPTEMBER SESSIONS

First Session

Written exam: Thursday 06/09/18, 15.00, Aula 2

Oral exam: Tuesday 11/09/18, 10.00, Aula 9

Second Session

Written exam: Tuesday 18/09/18, 15.00, Aula 2

Oral test: Friday 21/09/18, 10.00, Aula 9

SUMMER SESSIONS

First Session

Written exam: Monday 18/06/18, 15.00

Oral exam: Thursday 21/06/18, 10.00

Second Session

Monday 02/07/18, 15.00

Oral exam: Thursday 05/07/18, 10.00

General informations

Course timetable: Monday 16.00-17.45, Thursday 9.30-11.15, Friday 14.00-15.45

All lectures are in Aula 7

Tutoring timetable: Friday 9.30-11.15 (first lecture: 23/3/18)

Office hours: by appointment (please send an e-mail)

Textbooks:

• T. Apostol, Calculus, Vol. I (Chapters 12-16), Vol. II (Chapters 3-5), Second edition, Wiley

• David Lay, Linear algebra and its applications

• Massimo Gobbino, notes and video lectures of a Linear Algebra course (but in italian!)

Exercises:

• John Erdman's Exercises and Problems in Linear Algebra

• Robert Beezer's Exercise and Solution manual for a first course in Linear Algebra

Instructions for the exams

There will be a written and an oral exam. One is admitted to the oral exam only if the written exam is sufficient (at least 18/30).

The written exam consists on a number of exercises and is based on what was done during the course: theory and exercises.

The exercises given during the course and those left for the students are considered part of the course.

Definitions and simple proofs may be asked at the exam.

No books, notes, calculators or phones will be allowed during the exams.

Past exams

These are written exams for a similar course given by Prof. Iannuzzi in the last years. They are in italian.

Past exams

This is a written exam translated to english.

Past exam in english

Outline of the course

• Vectors in the plane R^2, scalar product, equations of lines, linear systems, matrices, Gauss algorithm.

• Vector in the space R^3 and in R^n. Scalar product, lines and planes in the space.

• Vector spaces, bases and dimension.

• Linear maps, matrices and linear systems revisited.

• Determinants.

• Scalar products, orthonormal bases and Gram-Schmidt. Orthogonal matrices.

• Diagonalization, eigenvalues and eigenvectors.

• Symmetric matrices and spectral theorem.

• Conics and quadrics.

Lecture diary - first half (taught by Arosio)

• Lecture 1, 8/3/18: Analytic geometry in the plane, operations on vectors, scalar product, angles, norm and distance of vectors, Cauchy-Schwarz inequality, Carnot's theorem. Lines in the plane, cartesian and parametric equations for a line.

• Lecture 2, 9/3/18: Geometric interpretation of cartesian equations, mutual position of 2 lines in the plane, passing from cartesian to parametric equations and viceversa. General facts about linear systems, first examples of Gaussian elimination.

• Lecture 3, 12/3/18: Gaussian elimination, coefficient matrix of a system, matrix in echelon form, pivots and free parameters, solution of a linear system. Analytic geometry in R^n, scalar product, cartesian and parametric equation of a plane in R^3.

• Lecture 4, 15/3/18: Cartesian equations of a plane through 3 points, mutual position of 2 planes, parametric and cartesian equations of a line in space, intersection of 2 lines in cartesian and in parametric form, mutual position of 2 lines in space, passing from cartesian to parametric equation and viceversa, orthogonal projection of a point to a line.

• Lecture 5, 16/3/18: Definition of field and of vector space over a field, subspaces, linear combinations, span, generators, linear independence.

• Lecture 6, 19/3/18: Bases, coordinates, linear systems and linear combinations, proof of the theorem of existence of a basis.

• Lecture 7, 23/3/18: Properties of bases, substitution lemma (without proof), from every set of generators we can extract a basis, every set of linearly independent vectors can be completed to a basis, consequeces. Sum and intersection of subspaces, proof of the Grassmann formula.

• Lecture 8, 24/3/18: Direct sum of subspaces. Product of matrices, writing a linear system as a matricial equation.

• Lecture 9 , 26/3/18: Linear maps, kernel and image, a matrix defines a linear map whose image is generated by the columns vectors.

• Lecture 10, 29/3/18: Injectivity and kernel of a linear map. Proof of the rank-nullity theorem. Linear isomorphisms.

• Lecture 11, 5/4/18: Matrix associated with a linear map with respect to a basis of the domain and a basis of the codomain.

• Lecture 12, 6/4/18: Change of basis matrix. Linear systems in terms of linear maps. Definition of inverse matrix and how to compute it via Gaussian elimination.

• Lecture 13, 9/4/18: How to find the matrix associated with a linear map with respect to some basis using the change of basis matrix.

• Lecture 14, 12/4/18: Exercise: write the matrix associated with the projection to a subspace in direct sum in R^3 with respect to the canonical bases. Structure theorem for the solutions of a linear system.

• Lecture 15, 13/4/18: The solution of a linear system is an affine space. Rouché-Capelli thorem, Discussion about existence of solutions for a system with parameter. Properties of the determinant, determinant of a 2x2 matrix, geometrical interpretation as the aerea of the parallelogram generated by the two vectors.

• Lecture 16, 16/4/18: Determinant of a 3x3 matrix, Sarrus rule. Laplace expansion by columns and by rows. How the Gaussian elimination affects the determinant. Determinant of an upper triangular matrix. The Laplace expansion by columns gives the determinant (without proof). Proof that the Laplace expansion by row also gives the determinant.

• Lecture 17, 19/4/18: Gaussian elimination implies uniqueness of the determinant, proof by induction that the determinant of a matrix is equal to the determinant of the transpose matrix. All rows of a square matrix are linearly independent if and only if all columns are linearly independent. Proof of Binet theorem.

• Lecture 18, 20/4/18: Cramer rule, inverse of a matrix using determinants, finding perpendicular vectors using determinants. Proof that the column-rank is equal to the row-rank.

• Lecture 19, 24/4/18: Calculating the rank of a matrix using the minors of the matrix. Linear systems with parameters.

• Lecture 20, 26/4/18: Orthogonal and orthonormal basis. Gram-Schmidt process. Orthogonal of a subspace and its properties.

• Lecture 21, 27/4/18: Exercise: find orthonormal bases of a subspace of R^n and its orthogonal. Orthogonal matrices, properties of transposed matrices. Orthogonal projection onto a subspace, and how to find its matrix in the canonical basis using change of basis matrix.

Lecture diary - second half (taught by Iannuzzi)

• First week: Planes in R^3, from parametric equations to the analytic equation and "viceversa". Sheafs of parallel planes. The vector product in R^3, properties and applications: the right/left hand rule, the area of parallelograms/triangles, the volume of a parallelepiped. Lines in R^3, from parametric equations to analytic equations and "viceversa", using the vector product and the rank/determinant criterion (principio dei minori orlati).

• Second week: The sheaf of (non parallel) planes containing a line in R^3, mutual position of planes/lines and planes/lines in R^3 by studying the rank of the full system of linear equations defining the two affine subspaces. Distance between a point/line/plane and a point/line/plane in R^3. How to determine the unique line orthogonal and intersecting two non-orthogonal distinct lines in R^3. Circles and lines tangent to a circle in R^2, spheres and planes tangent to a sphere in R^3, radius of a circle given as the intersection of a sphere and a plane. Matrixes associated to linear maps, basic formulas for the composition of maps and for the change of basis, projections and reflections, examples.

• Third week: Kernel and Image of a linear map rivisited in terms of a representative matrix, the rank of a linear map, the dimension theorem rivisited in terms of a representative matrix. Isomorphisms: equivalent conditions for L:V-->W to be an isomorphism in the finite dimensional case n = dim V = dim W, extensions and restiction of linear maps: a linear map is uniquely defined by freely assigning the image of each element of a chosen basis, examples. Linear operators, diagonalizable linear operators vs. diagonalizable matrixes, eigenvalues, eigenvector and eigenspaces, the cases of projections, reflections and rotations in R^2, the characteristic polynomial, geometric and algebraic multiplicity of an eigenvalue, examples where such multiplicities do/do not coincide.

• Fourth week: Complex numbers, polar form for the product, definition of complex exponential, statement of the fundamental Theorem of algebra, real polynomials have real and complex conjugate roots. The characteristic polynomial of A is equal to the characteristic polynomial of MAM^{-1}, the geometric multiplicity is smaller or equal than the algebraic multiplicity. Eigenspaces associated to distinct egenvalues are in direct sum. As a consequence, if there are n distinct eigenvalues, with n the dimesion of V, then L is diagonalizable. Equivalent conditions for L to be diagonalizable in terms of the dimensions of all eigenspaces. L is diagonalizable iff all roots of the characteristic polynomial are in the field and the algebraic multiplicity of each root coincides with the geometric one. Examples, rotations in R^2 and their complexifications in C^2, upper triangular matrixes, .... The coefficients of the characteristic polynomial: the trace and the determinant. In a closed field the trace is the sum off all eigenvalues, counting multiplicity, and the determinant is their product.

• Fifth week: Symmetric (self-adjoint) operators: characterization in terms of representing matrixes orthonormal basis. Spectral theorem for symmetric operators: proof in dimension 2 and 3. Consequences for symmetric matrixes. The standard Hermitian product, norm and distance, orthonormal basis. Unitary matrixes and unitary operators: characterization in terms of representing matrixes with respect to an orthonormal basis. Spectral theorem for symmetric operators (without proof). Consequences for unitary matrixes. Hermitian (self-adjoint) operators: characterization in terms of representing matrixes orthonormal basis. Spectral theorem for symmetric operators (with proof) and its consequences. Consequences for symmetric matrixes. Example.

• Sixth week: Locus of zeros Z_f of a real function f defined on R^n: Z_{tf} = Z_f for every non zero real number. Given a bijection F:R^n-->R^n the image F(Z_f) of Z_f is the locus of zeros of the composition of f with F^{-1}. Euclidean conics: the locus of zeros of a polynomial of degree 2. How to obtain a canonical form by diagonalizing the quadratic form and by completion of squares. Euclidean classification. The complete matrix and the matrix of the quadratic form: how to classify the conic by looking at invariants. Conics with center: the reflection with respect to the center leaves the conic invariant. Ellipses, hyperbolas and parabola as geometric locuses in E^2: focuses, directrixes, eccentricity, simmetries and different parametrizations.