Linear Algebra

obligatory courses

Level: Bachelor.

Classes are devoted to the presentation of the basic concepts of linear algebra. These are, among others: systems of linear equations and methods of solving them, linear spaces, base and dimension, linear transformations, the determinant of a matrix, eigenvalues ​​and eigenvectors, diagonalisation of a matrix, the scalar product and quadratic forms. In addition to mastering the techniques of linear algebra aim of the course is to develop student's ability to accurately and logical reason and prepare them for the applications of linear algebra in economics.

1 Systems of linear equations: solutions and general solutions, matrices, elementary matrix operations, solving the system of equations using Gaussian elimination.

2 Linear (or vector) spaces: examples, linear subspaces, linear combinations of vectors, linear independence, basis and dimension of a linear space, the coordinates of the vector in a given basis.

3 Linear transformations: examples , matrix representation of linear transformations, the algebra of linear transformations and matrix operations, matrix algebra .

4 Determinants: properties of determinants and methods of calculation.

5 Matrix Inverse and methods of finding the inverse matrix.

6 Applications determinant and rank of a matrix to solve linear equations : Kronecker - Capelli theorem and Cramer.

7 Vectors and eigenvalues ​​of linear transformations: Find the eigenvalues, the characteristic polynomial, bases of eigenspaces and diagonalisation of matrices.

8 Applications matrix diagonalisation.

9 Affine subspaces (or layers) of linear spaces, equations of the line and plane.

10 The standard scalar product: vector length, magnitude of vectors, orthogonal bases and orthonormal bases and the Gram-Schmidt procedure.

11 Quadratic forms: examples of matrix quadratic forms, Sylvester criterion of positive definiteness and tests of semidefinitness using eigenvalves.

Linear Algebra, K.M. Hoffman and R. Kunze, Pearson; 2 edition (April 25, 1971)

The ability to understand and use linear algebra in statistics, econometrics and mathematical models of decision making. Basic techniques of linear algebra, including: solving systems of linear equations, finding bases and dimensions of space, calculating rows, determinants and matrix inverse, finding the eigenvectors of linear transformations, diagonalization, testing positive (negative) definiteness of quadratic forms.

KU04, KW01

Evaluation of the course is via a written examination.