Acceptance of concepts and methods in linear algebra on more advanced level.

1. list basic notions of linear algebra
2. describe basic notions and results of linear algebra
3. derive basic results of linear algebra
4. explain the connection between linear algebra and problems of stability
5. describe the properties of matrix norm
6. convert a linear system of differential equations into a matrix form

the lectures include auditory exercises

five homeworks

included in the lectures

twice per week

matrix transformations of the plane http://www.zpm.fer.hr/~darko/la/linal.html

1. acquainting with the content, vector spaces, linear dependence and independence, linear hull, base and dimension, general vector spaces, vector spaces over finite fields, vector space of polynomials, spaces of functions, examples and exercises
2. intersection and sum of subspaces, direct sum, change of base, inner product space, inner product, orthogonality, orthogonal complement, orthogonal projection, decomposition of vector with respect to orthogonal base, Cauchy-Schwarz-Bunjakowski inequality, normed vector space, Hoelder inequality, Minkowski inequality, Eucliedan norm, p-norm, examples and exercises
3. Hilbert space, Banach space, l_p space, Lebesgue space, linear operators, matrix representation of linear operators, symmetry operator, roatation operator, defining linear operator on a base, change of base in representing lienar operators, similar matrices, examples and exercises
4. linear operators in the plane and in R^3, homothety operator, sqew operator, orthogonal projection onto a line, symmetry operator with resepct to a line, operator of rotation in the spaces, algebra of linear operators, examples and exercises
5. kernel and image of an operator, rang and defect and their connectionb, injectivity of linear operator, regular or isomorphic operators, isomorphic vector spaces, eighevectors and egenvalues, spectrum of a matrix, characteristic polynomial, spectral radius, diagonalization of operators, examples and exercises
6. properties of the upper triangular matrices, Schur's theorem, spectral mapping theorem, matrix polynomial, Hamilton-Cayley theorem, nilpotent matrices, polynomial of a nilpotent matrix, examples and exercises
7. Jordan form of a matrix, scalar product and diagonalization, Gramm-Schmidt orthogonalization, QR decomposition of a matrix, filling up the orthogonal set of vectors to orthogonal base, symmetric matrices, hermitian (self-adjoint) matrices, eigenvalues of symmetric matrices, examples and exercises
8. eigenvectors of symmetric matrices, orthogonal matrices and their properties, unitary matrices, spectral theorem of symmetric matrices, diagonalization of symmetric matrces, examples and exercises
9. midterm exam
10. matrix norm, operator norm, bounded linear operators, linear functionals, dual space, Riesz theorem about representation of linear functionals, convergence of matrices, series of matrices, exponential function of a matrix, function of a matrix, examples and excercises
11. spectral radius and the Neumenn series, spectra norm of a matrix, resolvent set, positive semidefinite matrix, positive definite matrix, definition of stabile matrices via the spectrum, definition of stability via the exponential function of a matrix, examples and exercises
12. Raus-Hurwitz stability criterion, stability of linear systems, Gershgorin theorem about circles, diagonally dominant matrices, Bauer's theorem about Cassini ovals, examples and exercises,
13. fixed point of a mapping, methodology of iterative methods, Jacobi's and Gauss-Seidel method, characterization of convergence of iterative methods, error analysis, examples and exercises
14. matrix analysis of differential equations, linear Cauchy problem, fundamental matrix, explicite solution of matrix linear differential equations, discrete dynamical systems in the plane, Frechet derivative, examples and exercises
15. singular decomposition of matrices, trace of a matrix, diagonal rectangular matrix, singular values of a matrix, polar decomposition of a matrix, pseudoinverse matrix, minimization of |Ax-b|, examples and exercises