Syllabus
1 Systems of linear equations, Gauss elimination method. 2 Matrix operations, inversion of a matrix. 3 Groups, vector spaces. Subspaces, linear independence, linear span. 4 Basis, dimension, Steinitz theorem. 5 Rank of a matrix, Frobenius theorem. 6 Linear maps and their matrices, kernel and image, rank-nullity theorem. 7 Coordinates and their transformations, similarity of matrices, trace of a matrix and of a linear map. 8 Scalar product, Cauchy-Schwarz inequality. 9 Orthogonal complement, orthogonal projection. 10 Permutation and its sign. 11 Determinant and its properties.
Expansion along a row and a column. 12 Determinant of a product, inverse matrix formula, Cramer's rule. 13 Eigenvectors and eigenspaces. 14 Block matrices, sum and direct sum of subspaces.
Annotation
This course gives, together with parallel courses on analysis, a basic course of mathematics for physicists. Emphasis is given also to relationship of all these disciplines.
Keywords linear spaces, dimension, matrices, determinants, groups and algebras of matrices, eigenvalues,
Jordan normal form.