58416 - Linear Algebra and Geometry

Learning outcomes

Fundamental notions in linear algebra and their applications to analytic geometry.

Course contents

The program is divided into teaching units. Each unit is treated through traditional lessons and exercises, discussions with students. Each concept is introduced by definition, examples and counterexamples. A proof of every statement is provided, except in special cases.

1. Operations with sets: union, intersection, difference, Cartesian product. Equivalence relations, equivalence classes, partitions of a set. Applications; injective, surjective, bijective applications; composition of applications, reverse application. Natural, integer, rational, real, complex numbers. Every polynomial equation has solution in the complex numbers (without proof).

2. Groups, subgroups; hints on homomorphisms and isomorphisms of groups. Examples: (Z, +), (Q * ,.), integers modulo a given integer, symmetric groups. Fields: rational, real, complex; hints to finite fields with a prime number of elements. Vector spaces and their subspaces. Examples: n-ples of elements of a field, polynomials with coefficients in a field, functions on a finite set with values in a field. Counterexamples: curves, lattices, cones, union of subspaces. The intersection of subspaces is a subspace.

3. 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.

4. All the bases of a vector space have the same cardinality. 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). Direct sum, Grassmann formula. Two ways of describing a vector subspace: parametric form and Cartesian form; dimension.

 

5. Linear applications. The composition of linear applications is linear. Vector space of linear applications between two given vector spaces. Kernel and image of a linear application; the kernel and the image are subspaces. Link with injectivity and surjectivity. Rank-nullity theorem. Isomorphism; being isomorphic is an equivalence relationship between vector spaces.

6. 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. A linear application is an isomorphism if and only if it sends bases to bases. All vector spaces of dimension n on a given field are isomorphic to each other.

7. Identity matrix, invertible matrices. A square matrix is invertible if and only if its column vectors are linearly independent; rank of a matrix. Linear general group, linear special group. Determinant of a square matrix: recursive definition and its properties (without proof). Formula for the inverse matrix of a given matrix. Matrix of a change of basis. Similarity; two matrices are similar if and only if they represent the same linear map. Square matrices representing the same linear map in different bases have the same determinant; determinant of a linear application.

8. Solution of n × n systems by inversion of the coefficient matrix ("Cramer's method"). The set of solutions of a homogeneous linear system is a vector subspace. Affine spaces and their subspaces. Parametric representation and Cartesian representation of an affine subspace. The set of solutions of a linear system, if it is not empty, is an affine subspace of dimension n-rk A. Gauss-Jordan elimination method. Applications to geometry: straight lines and planes passing through data points; intersections of lines and planes; parallelism.

9. 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. Eigenspaces and their properties. Bases of eigenvectors. Algebraic and geometric multiplicity. 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. Example: the derivative application on spaces of polynomials or trigonometric functions; hints to the complex exponential. Nilpotent applications; a non-zero nilpotent linear map is not diagonalizable. Jordan canonical form (without proof).

10. Bilinear forms. Bijection between bilinear forms and matrices (in a given basis). Bilinear symmetric and antisymmetric forms, symmetric and antisymmetric matrices. Every matrix is written in a unique way as the sum of a symmetric matrix and an antisymmetric matrix. 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 (ie every symmetric matrix is congruent to a diagonal matrix).

11. Canonical form of a real bilinear form; signature. Sylvester's theorem, i.e. the signature does not depend on the chosen basis. 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.

 

12 Scalar products. Examples of scalar products: the standard scalar product of n-ple of elements of a field, the 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. Convex angle between two vectors. Norm of a vector and its properties. Property that must satisfy a function on a set to be called a "distance". Discrete distance, Manhattan distance, Euclidean distance. Sets of orthogonal and orthonormal vectors. An orthogonal set of vectors is linearly independent. 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. Spectral theorem (hints of proof). Subspace orthogonal to a given subspace. Applications to geometry: affine subspace orthogonal to a given subspace and passing through data points.

 

13. 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. Orthogonal matrices, orthogonal group, orthogonal special group. 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. A linear application is an isometry if and only its matrix with respect to any orthonormal basis is orthogonal. Determinant and eigenvalues of an isometry. Classification of isometries in dimension 2: rotations and symmetries; dihedral groups. Classification of isometries in dimension 3. Outline of the dual of a vector space: isomorphism between V and V * given by the choice of a base or of a non-degenerate bilinear form; canonical isomorphism between V and V **. Hermitian products. Notes on unitary applications and unitary matrices.

 

Readings/Bibliography

B. Martelli, Geometria e Algebra https://people.dm.unipi.it/martelli/Alg%20Lin.pdf

- M. Manetti, Algebra Lineare per Matematici,

https://www1.mat.uniroma1.it/people/manetti/AL2017/algebralineare.pdf

- S. Lang, Linear Algebra, Springer

Teaching methods

Traditional lessons and exercises. Several hours will be dedicated to discussions with students both in class or in small groups. A tutor will support the students with exercises.

Assessment methods

The exam is written and oral. The written exam consists of some exercises.This writenn part can be divided into three parts for students attending lessons.

Teaching tools

See the preceding items

Office hours

See the website of Nicoletta Cantarini