Linear Algebra

Objective

We present the fundamental concepts of linear and bilinear algebra, emphasising new concepts and their applications in economics. Particular attention is given to matrix reduction, and especially the real symmetrical case. Finally we underline the importance of the orthogonal projection concept and its applications.

Assessment:

The overall grade for the course will be the average of the continuous assessment (“contrôle continu”, CC) grade (50%) and the written final exam (50%).

The continuous assessment grade is made up of three elements, each graded out of twenty points: (i) the mid-term grade, (ii) the grade for attendance at tutorial sessions (TD), whose attendance is mandatory, and (iii) the grade for participation in tutorial sessions. The CC grade is calculated as follows: 50% of the mid-term grade + 25% of the attendance grade + 25% of the maximum between the participation grade and the mid-term grade.

The attendance grade is calculated according to the scale available on the school's intranet.

Planning

1. Vector spaces- Vector spaces. Vector sub-spaces. Linear independence, span, bases and dimension. Kernel, image and rank of a linear map. Direct sums of vector sub-spaces and associated projectors.
2. Matrices -Endomorphism matrix in a given basis. Changes of basis. Matrix calculations. Equivalent and similar matrices, characterisations.
3. Determinant -Determinant of a family of vectors in a given basis, determinant of an endomorphism, determinant of a square matrix. Calculation methods. Inverting a square matrix using the cofactor method
4. Inverses -Using elementary operations on rows and columns to calculate rank or invert a matrix and to resolve a linear system.
5. Endomorphism reduction -Diagonalisation of endomorphisms and square matrices. Proper values and proper vectors, characteristic polynomial. Characterisations of diagonalisable endormorphisms. Spectral theorem.
6. Applications of reduction -Application of the reduction of square matrices to power calculation and the resolution of equations and linear recurrent or differential systems.
7. Quadratic forms -Definitions, orthogonality, associated matrix in a given basis. Changes of basis. Congruent matrices. Gauss reduction. Application to the search for an orthogonal basis and a quadratic form signature.
8. Euclidean spaces -Definitions, examples. Schmidt orthogonalisation. Orthogonal matrices, properties. Reduction of orthogonal matrices. Orthogonal projection on a vector sub-space. Calculating the distance to a vector sub-space. Different expressions. Applications.
9. Real symmetrical matrices -Real symmetrical matrices. Symmetrical endomorphisms. Diagonalisation of a real symmetrical matrix in an orthonormed basis. Simultaneous reductions.