Faculty of Applied Mathematics & Control Processes, St.Petersburg State University
Lecturer: Prof. Alexei Uteshev
Binomial theorem. Pascal's triangle.
The greatest common divisor (GCD). Euclidean algorithm. Linear representation for GCD, the continuant.
Divisibility. Coprime numbers. Prime numbers. Factorization.
Euler's totient function.
Congruence. Modular arithmetic. Square-and-multiply algorithm. Theorems by Fermat and Euler.
Linear congruences solving. Modular inverse. Wilson's theorem. Chinese remainder theorem.
De Moivre's formula. Trigonometric equalities.
Root extraction. Roots of unity. Primitive roots.
Basic notions, Horner's process.
The fundamental theorem of Algebra (formulation). Viète's formulas. Factorization. Multiplicity of the root.
Solving by radicals: cubic and quartic equations. Cardano's formula for the cubic equation with real coefficients: casus irreducibilis.
Divisibility. The greatest common divisor. Euclidean algorithm. Relatively prime polynomials, Bézout's identity.
Multiple root finding.
Polynomials with real coefficients. Bounds for the roots: by Maclaurin, Lagrange and Newton. Descartes rule of signs.
Numerical methods for solving algebraic equations: the Ruffini-Horner method, the Lagrange (continued fractions) method, Newton's iterations.
Multivariate polynomials, their representations. Taylor's expansion, extrema conditions. Algebraic equations.
Systems of linear equations. Gaussian elimination.
Matrices and their elementary properties.
Definition of the determinant. Permutations.
Fundamental properties of the determinant.
Minors and cofactors. Expansion of the determinant.
The Binet-Cauchy theorem. Cauchy's inequality.
Special types of determinants (Vandermonde, Hankel, with integer entries, characteristic polynomial)
Conditions for the consistency of linear system. The Kronecker-Capelli theorem. Complete solution. Homogeneous linear equations. The fundamental system of solutions.
Polynomial interpolation: methods by Lagrange and Newton.
The least squares method.
Overdetermined system of equations. Pseudoinverse.
Resultant and its linear representation.
Elimination procedure for the system of bivariate equations. Bézout's theorem (on the number of solutions in $ \mathbb C^2 $).
The fundamental theorem of algebra (proof).
Basic notions. Canonical form. Lagrange's method
Sylvester's law of inertia. Equivalence of quadratic forms.
Sign-definiteness of quadratic forms. Sylvester's criterion.
Hankel matrices in the problem of root localization: Newton's sums, theorems by Jacobi and Joachimsthal.
Roots in special domains of the complex plane: criteria for the stability and discrete stability of a polynomial.
Binary operation. Group, subgroup. Generators. Lagrange's theorem.
Multiplication (Cayley's) table. Factor group. Isomorphism.
Rings, fields, algebras. Quaternions.
Basic notions. Isomorphism.
Linear dependency, basis. Relative linear dependency. Linear homogeneous system of equations and its fundamental system of solutions
Sum and intersection of subspaces. Linear inhomogeneous system of equations.
Direct sum of subspaces
Change of basis matrix
Orthogonal basis. The Gram-Schmidt orthogonalization process.
Gram's determinant. Nonnegativity, distance evaluation, Hadamard's inequality for the determinants, volumes
Kernel and image
Matrix of a linear mapping
Invariant subspaces. Eigenvectors and eigenvalues of an operator. Characteristic polynomial. Non-derogatory matrices.
Properties of a characteristic polynomial. The Cayley-Hamilton theorem.
Symmetric matrix and properties of its eigenvalues and eigenvectors.
Generalized eigenvectors. Jordan canonical form (JCF).
Linear difference equation
Probability theory. Gambler's ruin problem. Markov chain.
Analytic functions of a matrix. Matrix norm. Linear ordinary differential equations. Notion of stability
Characteristic polynomial evaluation. Leverrier's method. Krylov's method.
Partial eigenvalue problem
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3. Horn R.A.,Johnson C.R. Matrix analysis. Cambridge University Press. 1986
4. Strang G. Linear Algebra and its Applications. Cengage learning. 2006
5. Uspensky J.V. Theory of Equations. McGraw-Hill. 1948