09757 - Geometry and Algebra

Learning outcomes

At the end of the course, the student will have basic knowledge of matrix and vector calculus. Specifically, the student will be able to: solve determinants, inverse matrix and linear systems; calculate eigenvalues, eigenvectors and eigenspaces of endomorphisms and matrix.

Course contents

The course is made up of 20 lessons of 3 hours, divided into the 8 teaching units described below.

1. Recalls on sets. Equivalence relations. Applications; injective, surjective, biunivocal applications; composition of applications, reverse application. Natural, integer, rational, real, complex numbers. Vector spaces and their subspaces. Examples: n-ples of elements of a field, polynomials with coefficients in a field, functions with values in a field. Counterexamples: curves, lattices, cones, union of subspaces. The intersection of subspaces is a subspace. Linear combinations, subspace generated by a set of vectors. Generating sets and linearly independent sets. Bases. A set of vectors is a basis if and only if each vector is written uniquely as a linear combination of its elements. Coordinates of a vector in a given base. To complete to a base, to extract a base. [3 lessons]


2. All the bases of a vector space have the same cardinality (without proof). Dimension. Canonical basis for the previous examples (n-ple of elements of a field, polynomials with coefficients in a field, functions on a finite set with values in a field). Sum of subspaces, direct sum, Grassman's formula (without proof). Two ways of describing a vector subspace: parametric form and Cartesian form; links with dimension. [2 lessons]


3. Linear applications; examples. The vector space of linear maps between two data spaces. The composition of linear applications is linear, the inverse of a linear application is linear. Core and image of a linear application; the core and the image are subspaces. Link with injectivity and surjectivity; isomorphisms. Rank theorem. There is one and only one linear application that takes given values on a given basis. Matrix of a linear map in given bases; isomorphisms between the vector space of linear maps between two given vector spaces and the vector space of m x n matrices. The composition matrix of two applications is the "row-by-column product" of the two corresponding matrices (proof only in the 2x2 case). A linear application is an isomorphism if and only if it sends bases to bases (optional proof). All vector spaces of dimension n on a given field are isomorphic to each other. [3 lessons]


4. Identity matrix, invertible matrices. A square matrix is invertible if and only if its column vectors are linearly independent; rank of a matrix. Basic changes; examples. Similarity; two matrices are similar if and only if they represent the same linear map (optional proof). Determinant of a square matrix: recursive definition and its properties (without proof). Formula for the inverse matrix of a given matrix. Square matrices representing the same linear map in different bases have the same determinant; determinant of a linear application. Vectorial Product. [2 lessons]

5. Gauss-Jordan elimination method. The set of solutions of a homogeneous linear system is a vector subspace. Affine subspaces, their parametric and Cartesian representations. The set of solutions of a linear system, if it is not empty, is an affine subspace of dimension n-rk A. Rouché-Capelli's theorem. Examples of linear systems on spaces of functions. Applications to geometry: straight lines and planes passing through data points; intersections of lines and planes; parallelism. Examples and exercises. Review. [2 lessons]


6. Diagonal matrices and their properties. Eigenvalues and eigenvectors of a linear map. Characteristic polynomial; examples of linear applications that have no eigenvalues in the rational field or in the real field. Autospaces. Bases of eigenvectors. Algebraic and geometric multiplicity. Inequality between algebraic and geometric multiplicity (optional proof). An application is diagonalizable on a given field if and only if all the eigenvalues belong to the field and the algebraic multiplicity of each eigenvalue is equal to its geometric multiplicity (optional proof). Nilpotent applications; a non-zero nilpotent linear map is not diagonalizable. Outline of Jordan's canonical form (without proof). Example: the derivation of polynomials. [2 lessons]


7. Bilinear forms. Bjection between bilinear forms and matrices (in a given base). Bilinear symmetric and antisymmetric forms, symmetrical and antisymmetric matrices. Transpose of a matrix and its properties. Congruence between matrices; two matrices are congruent if and only if they represent the same bilinear form. Diagonalization of bilinear forms. For every symmetric bilinear form there is a diagonalizing base, that is, every symmetric matrix is congruent to a diagonal matrix (hints of proof). Spectral theorem (without proof). Canonical form of a real bilinear form; signature. Sylvester's theorem, i.e. the signature does not depend on the chosen basis (without proof). Canonical form of a complex bilinear form; rank. Quadratic forms. Correspondence between quadratic forms and symmetrical bilinear forms. Real quadratic forms positive and negative definite, positive and negative semidefinite, indefinite; their signature. [3 lessons]


8. Scalar products. Notable examples of scalar products: standard scalar product of n-ple of elements of a field, standard scalar product of functions (with values of a field) on a finite set, integral of the product of continuous functions on a closed and bounded interval. Cauchy-Schwatz inequality (without proof). Convex angle between two vectors. Norm of a vector and its properties. Euclidean distance and its properties. Examples of other functions that verify the same properties: discrete distance and Manhattan distance. Sets of orthogonal and orthonormal vectors. An orthogonal set of vectors is linearly independent. Subspace orthogonal to a given subspace. Existence of orthonormal bases. The dot product of two vectors is equal to the standard dot product of their coordinates with respect to an orthonormal basis. Applications to geometry: affine subspace orthogonal to a given subspace and passing through data points. Isometries of a vector space (with respect to a given scalar product). A linear map is an isometry if and only if it preserves the norm of each vector. Each isometry is an isomorphism. Each isometry preserves the angles. A basis is orthonormal if and only if the matrix of the change of basis, with respect to a given orthonormal basis, is orthogonal. A linear application is an isometry if and only if it sends orthonormal bases to orthonormal bases (optional proof). A linear application is an isometry if and only its matrix with respect to any orthonormal basis is orthogonal (optional proof). Determinant and eigenvalues of an isometry. Classification of isometries in dimension 2: rotations and symmetries. Outline of the classification of isometries in dimension 3. [3 lessons]

Readings/Bibliography

Students may choose one of the two following textbooks:


- Marco Manetti, Linear Algebra for mathematicians, downloadable for free on


https://www1.mat.uniroma1.it/people/manetti/dispense/algebralineare.pdf


- A. Bernardi and A. Gimigliano, "Linear algebra and analytic geometry", Citta'Studi edizioni.


In any case, it is advisable to integrate the textbook with the course notes.


Exercise sheets are available on the Virtual page of the course.


Further exercises can be found on:


Anichini-Conti-Paoletti, Algebra and Linear and Analytical Geometry EXERCISES AND PROBLEMS, second ed, Pearson.

Teaching methods

The course consists of 60 hours of lesson, during which the topics will also be presented through definitions, theorems, demonstrations, examples, counterexamples and exercises. The solution of exercises of various levels of difficulty will be illustrated to the students and exercises to be solved independently will be proposed to them. In addition to the 60 hours of lessons, some hours of tutoring will be held dedicated to correcting these exercises and clarifying doubts.

Assessment methods

It is highly recommended, although not mandatory:

- attending the lessons regularly, participating proactively (e.g. with questions);

- carrying out and uploading the assigned exercises;

- taking advantage of the office hours to clarify any doubts in time.

The exam consists of a written and an oral test. The written test is divided into 10 multiple choice questions and two exercises, divided into various subproblems. Those who pass the written test are admitted to sit the oral test. During the oral exam the student will have to prove that he has understood the fundamental concepts of the course, and be able to apply them to examples and problems; he will have to know precisely the definitions, the statements of the theorems and, possibly, their proof.

The dates of the exams will be published well in advance on almaesami or on the website of the degree course.

Teaching tools

- Recommended textbooks and workbooks

- Lessons that can be followed in person or (if required by the epidemiological situation) online

- Course page on Virtual, with the program carried out, the assigned exercise sheets, announcements, etc.

-Almaesami and EOL apps.

Office hours

See the website of Luca Moci