● school profile, Shandong University
Founded in 190 1, Shandong University is known as the earliest modern higher education university in China. Its medical discipline originated from 1864, which opened modern higher medical education in China. Since its birth, the school has experienced several historical development periods, such as Shandong University Hall, National Qingdao University, National Shandong University, Shandong University and the new Shandong University formed by the merger of former Shandong University, Shandong Medical University and Shandong Polytechnic University. Since 120, Shandong University has always adhered to the mission of "storing talents for the world and enriching Qiang Bing", deeply practiced the spirit of "endless learning and lofty aspirations", made unremitting efforts, passed down from generation to generation, accumulated and formed the school spirit of "advocating truth, seeking truth from facts and being innovative", trained more than 600,000 talents of all kinds, and made contributions to national and regional economic and social development.
● The nature, objectives and contents of the examination of 825 "Linear Algebra and Ordinary Differential Equations".
First, check the target.
Linear algebra and ordinary differential equations are examination subjects with selective function, which are set up for recruiting graduate students from science colleges and mathematics colleges. Its purpose is to scientifically, fairly and effectively test whether candidates have the basic quality, general ability and training potential necessary for studying for a master's degree in mathematics, so as to select outstanding talents with development potential for admission. Its main purpose is to test candidates' mastery of linear algebra and ordinary differential equations and their ability to solve problems by using relevant knowledge. Candidates are required to systematically understand the basic concepts and theories of linear algebra and ordinary differential equations, and master the basic methods of linear algebra and ordinary differential equation theory. Candidates are required to have abstract thinking ability, logical reasoning ability, spatial imagination ability, calculation ability and the ability to comprehensively apply what they have learned to analyze and solve problems.
Second, the examination form and examination paper structure
1. Full score of test paper and test time
The full mark of the test paper is 150, and the test time is 180 minutes.
answer the question
The answer methods are closed book and written test.
3. Problem structure
The topics are calculation questions and proof questions.
Third, the examination content and requirements
Ⅰ. Ordinary differential equations
Some basic concepts of 1. differential equation
(1) exam content
1) ordinary differential equation
2) Order
3) Linearity and nonlinearity
4) Solutions, implicit solutions, general solutions and special solutions.
(2) Examination requirements
1) Understand the relationship between differential equations and some practical problems in the objective world.
2) Master the basic concepts of linear and nonlinear, general solution and special solution in differential equations.
3) Understand the geometric meaning of the first-order equation and its solution.
2. Elementary solution of first-order differential equation
(1) exam content
1) variable separation equation, homogeneous equation and equation that can be transformed into variable separation.
2) Linear equation, Bernoulli equation
3) The concept of the inherent equation, the necessary and sufficient conditions and the general solution of the inherent equation. The concept of integral factor and its solution
4) Solutions of first-order implicit equations (four kinds of equations)
(2) Examination requirements
1) can correctly identify the types of first-order equations.
2) Master the solution of variable separation equation, homogeneous equation and variable separation equation.
3) Master the solution of first-order linear equation and Bernoulli equation.
4) Master the basic methods of solving the eigenequation and finding the integral factor.
5) Master the solution of the first-order implicit equation.
3. Existence Theorem of First Order Differential Equation
(1) exam content
1) the existence and uniqueness theorem of solutions of first order differential equations; Approximate solution and error estimation
2) continuation theorems of solutions in bounded and unbounded domains
3) The continuous dependence and differentiability theorem of solutions on initial values.
4) The concept, solution and Clairow equation of odd solution.
(2) Examination requirements
1) Understand and master the existence and uniqueness theorem and its proof.
2) Will find the approximate solution of the equation and estimate its error.
3) Understand the continuation theorem of the solution
4) Understand the theorem of continuous dependence of solutions on initial values and the theorem of differentiability of solutions to initial values.
5) Understand the concept of singular solution and find the singular solution of the equation.
6) Mastering the solution of Clayro equation.
4. Higher order differential equations
(1) exam content
Properties and Structure of Solutions of Homogeneous Linear Equations (1)
2) The structure of general solution of nonhomogeneous linear equation and the method of constant variation.
3) Solving the general solution of homogeneous linear equation with constant coefficients,
4) Solving the special solution of non-homogeneous equation with constant coefficients.
5) order reduction of higher-order equations
(2) Examination requirements
1) to master the properties of solutions of homogeneous linear equations and the structure of general solutions.
2) Skillfully solve homogeneous and inhomogeneous linear equations with constant coefficients.
3) Will use the price reduction method to solve the higher-order equation.
5. Linear differential equations
(1) exam content
1) Existence and Uniqueness Theorem of First Order Linear Equation
2) General theory of linear equations
3) Standard basic solution matrix of linear equations with constant coefficients
4) Calculation of basic solution matrix
(2) Examination requirements
1) understand the existence and uniqueness theorem of first-order linear equations.
2) Understand the properties of solutions of linear equations.
3) Mastering the structure of general solution of linear equations, we can find the solution vector of non-homogeneous linear equations by constant variation method.
4) Find the basic solution matrix of linear equations with constant coefficients.
Ⅱ. Linear Algebra
1. Determinants
(1) exam content
Definition and basic properties of 1) determinant
2) Calculation of determinant
3) The determinant is expanded by rows (columns)
(2) Examination requirements
1) Understand the concept of determinant and use its properties to calculate determinant.
2) Will use Cramer's rule to solve linear equations.
3) Master the application of determinant in rows (columns)
2. Linear equation
(1) exam content
1) linear correlation (uncorrelated), rank of vector group
2) Rank of matrix
3) Basic solution system of homogeneous linear equations, general solution
4) Necessary and sufficient conditions, structures and general solutions of nonhomogeneous linear equations.
(2) Examination requirements
1) will discuss the linear correlation (uncorrelation) of vector groups and calculate the rank of the matrix.
2) Can calculate the basic solution system and general solution of homogeneous linear equations.
3) Grasp the necessary and sufficient conditions for the existence of solutions for nonhomogeneous linear equations, and calculate their general solutions.
4) Grasp the relationship between the basic solution system of homogeneous linear equations and matrix rank.
3.[ number] matrix
(1) exam content
Operation and properties of 1) matrix, inverse matrix of matrix
2) elementary transformation and elementary matrix
3) Rank sum determinant of product matrix
4) Application of Block Matrix
(2) Examination requirements
1) Understand and master the operations and properties of matrices.
2) Find the inverse matrix of the matrix.
3) Master the connection between elementary transformation and elementary matrix.
4) Master the application of block matrix.
4. Quadratic type
(1) exam content
The canonical form of 1) quadratic form and the contractual relationship of matrix.
2) Inertia theorem
3) Positive definite matrix and positive definite quadratic form
4) Semi-positive definite matrix and semi-positive definite quadratic form
(2) Examination requirements
1) master the solution of quadratic standard form.
2) Grasp the inertia theorem and its application.
3) Master positive definite matrix and positive definite quadratic form.
4) Understand semi-positive definite matrix and semi-positive definite quadratic form.
5. Linear space
(1) exam content
The basic concepts, bases and dimensions of 1) linear space.
2) subspace, subspace operation and dimension formula of linear space.
3) Direct sum decomposition of linear space and isomorphism of linear space
(2) Examination requirements
1) Master the basic concepts, bases and dimensions of linear space.
2) Master the operations of subspace and dimension formulas.
3) Master the direct sum decomposition of linear space.
6. Linear transformation
(1) exam content
1) linear transformation and matrix
2) Eigenvalues and eigenvectors, invariant subspaces
3) Characteristic polynomial and minimum polynomial of matrix.
4) Diagonalizable matrix
(2) Examination requirements
1) Grasp the correspondence between linear transformation and matrix.
2) Master the calculation of eigenvalues and eigenvectors.
3) Grasp the equivalent conditions of matrix diagonalization.
4) Understand the direct sum decomposition of linear space relative to a linear transformation and its application.
7.- Matrix?
(1) exam content
1) operation and equivalence of polynomial matrix, remainder division of polynomial matrix
2) Similar equivalence conditions of digital matrices.
3) Determinants, invariable factors and basic factors
4) Jordan canonical form and rational canonical form of matrix
(2) Examination requirements
1) Grasp the similar equivalence conditions of matrices.
2) Master the calculation of elementary factors, and be able to calculate Jordan standard form of matrix.
3) Master the relationship between the minimum polynomial of the matrix and the invariant factor.
4) Understand the rational canonical form of matrix.
8. European space
(1) exam content
The basic concept of 1) Euclidean space and the properties of inner product.
2) Standard orthogonal basis, orthogonal transformation and orthogonal matrix, symmetric transformation and symmetric matrix.
3) Eigenvalues and eigenvectors of real symmetric matrices
4) Principal axis problem of real quadratic form
(2) Examination requirements
1) Master the basic concepts of European space and the properties of inner product.
2) Master the similar canonical form of real symmetric matrix.
3) Master the properties of orthogonal matrix.
4) Understand the direct sum decomposition of Euclidean space about subspace.
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