2021 Ocean University of China 856 Advanced Algebra Postgraduate Examination Outline Text

856 Advanced Algebra

1、 Nature of examination
Advanced algebra is a professional basic course for the entrance examination of master's degree students in mathematics and statistics.
II investigate target
Try to reflect the characteristics of mathematics related master's degree, scientifically, accurately and normatively evaluate the basic quality and comprehensive ability of candidates for advanced algebra, specifically examine the candidates' mastery of the basic theory of advanced algebra, as well as their ability to analyze and solve problems using the theory and methods of advanced algebra.
Test the examinee's mastery and application ability of advanced algebra theory at three levels. The basic requirements of the three levels are:
1. Understand and master basic concepts and theories;
2. Ability to analyze, calculate and reason basic problems with basic theories;
3. Ability to comprehensively use advanced algebra knowledge to analyze and solve problems.
3、 Examination form
（1） Full score of examination paper and examination time
The full score of this test paper is 150, and the test time is 180 minutes.
（2） Answer method
The answer method is closed book and written examination.
（3） Test paper structure
(1) Score composition of test paper:
The polynomial theory part accounts for about 20 points;
The matrix theory part accounts for about 60 points;
The linear space theory part accounts for about 70 points.  
(2) Types of questions include: fill in the blanks, short answer questions, calculation questions, proof questions, etc.
4、 Examination content
（1） Polynomial theory
1. General theory of univariate polynomials
Concept, operation, derivative and basic properties;
2. Divisibility theory
The concept of division, the greatest common factor, the concept and nature of coprime;
3. Factorization theory
Irreducible polynomials, factorization, multiple factorization, factorization of real coefficient and complex coefficient polynomials, determination of irreducibility of rational coefficient polynomials, etc;
4. Root theory
Polynomial functions, roots of polynomials, rational roots of rational coefficient polynomials, relations between roots and coefficients, etc;
5. General Theory of Multivariate Polynomials
Concept of multivariate polynomial, symmetric polynomial.
（2） Matrix theory
1. Determinant Theory and Calculation
The concept, nature and calculation of determinant; Cramer's law, Laplace's theorem.
2. Linear equations
Linear correlation and independence of vectors and vector groups; The structure of solutions of linear equations.
3. Matrix
Various operations and operation rules of matrix, rank of matrix, inverse of matrix, corresponding operations and properties of block matrix.
4. Quadratic type
Basic concept of quadratic form, matching method and congruent transformation method to transform quadratic form into standard form, inertia theorem, positive definite, semi positive definite, semi negative definite quadratic form and matrix determination.
（3） Linear space theory
1. Linear space
The definition and properties of linear space; Linear correlation and relevant conclusions; Rank and maximal linear independence group; The basis and dimension of linear space; Base transformation and coordinate transformation formula; Linear subspace; Intersection, sum and direct sum of subspaces; Isomorphism of linear space.
2. Linear transformation
The definition of linear transformation and its basic properties; Linear transformation operation; Matrix of linear transformation; Similarity matrix; Eigenvalues and eigenvectors of matrices; Eigenvalues and eigenvectors of linear transformations; Hamilton Kelley theorem; Similarity diagonalization; Range and kernel of linear transformation; Invariant subspace; The matrix simplification of invariant subspace and linear transformation; Jordan canonical form; Minimum polynomial.
3. L matrix
L The concept of matrix; Equivalence of l matrix; The canonical form, invariant factor and determinant factor of matrix under elementary transformation; Elementary factors of l matrix; The method of finding the canonical form of l matrix; The necessary and sufficient conditions of matrix similarity; Jordan canonical form and rational canonical form of matrix.
4. Euclidean space
Inner product and Euclidean space; Length, included angle and orthogonality; Measurement matrix; Orthonormal basis; Orthogonal matrix; Isomorphism of Euclidean space; Orthogonal transformation; Orthogonal subspace and orthogonal complement; Canonical form of real symmetric matrix; Symmetry transformation; Distance from vector to subspace; Least squares method.
5、 Whether to use a calculator
No.
 

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