Syllabus
Determinants. Basic properties, determinant of a block matrix, the expansion of a determinant under a row and a column, the theorem on multiplication of determinants, adjugate matrix, inverse matrix, Cramer´s rule, rank of a matrix, calculation of determinants; examples.
Similarity, characteristic polynomial of a matrix, eigenvalues and eigenvectors, minimal polynomial of a matrix, Cayley-Hamilton theorem, similarity of matrices, simple Jordan matrix, Jordan matrix, the existence of the Jordan canonical form and the methods of evaluation, eigenvalues of symmetric matrix; examples.
Linear forms and dual space. Matrix and analytical expression of a linear form, dual space, dual basis; examples.
Bilinear forms. Matrix and analytical expression of a bilinear form, vertices, symmetric and antisymmetric forms, polar basis, quadratic forms, bilinear and quadratic form on real spaces, normal basis and normal expression, the law of inertia, signature, classification of forms; examples.
Annotation
Other parts of linear algebra following the subject Linear algebra II (determinants, similarity, linear forms, bilinear and quadratic forms, unitary spaces).