I. Determinants
Examination content
The concept and basic properties of determinant The expansion theorem of determinant by row (column)
Examination requirements
1. Understand the concept of determinant and master its properties.
2. The properties of determinant and determinant expansion theorem will be applied to calculate determinant.
Second, the matrix
Examination content
Concept of matrix, linear operation of matrix, multiplication of matrix, concept and properties of transposed inverse matrix of determinant matrix, necessary and sufficient conditions for matrix reversibility, equivalent block matrix of elementary transformation of matrix and rank matrix of elementary matrix and its operation.
Examination requirements
1. Understand the concepts and properties of matrix, identity matrix, quantitative matrix, diagonal matrix, triangular matrix, symmetric matrix and antisymmetric matrix.
2. Master the linear operation, multiplication, transposition and its operation rules of matrix, and understand the determinant properties of square matrix power and square matrix product.
3. Understand the concept of inverse matrix, grasp the properties of inverse matrix and the necessary and sufficient conditions of matrix reversibility, understand the concept of adjoint matrix, and use adjoint matrix to find inverse matrix.
4. Understand the concept of elementary transformation of matrix, understand the properties of elementary matrix and the concept of matrix equivalence, understand the concept of matrix rank, and master the method of finding matrix rank and inverse matrix by elementary transformation.
5. Understand the block matrix and its operation.
Third, the vector
Examination content
The linear combination of concept vectors of vectors and the linear representation of vector groups are linearly related to the largest linear independent group of linear independent vector groups. The relationship between the rank of equivalent vector group and the rank of matrix. Base transformation and coordinate transformation in vector space and their related concepts. Orthogonal normalization of inner product linear independent vector group defines orthogonal matrix of orthogonal basis and its properties.
Examination requirements
1. Understand the concepts of n-dimensional vectors, linear combinations of vectors and linear representations.
2. Understand the concepts of linear correlation and linear independence of vector groups, and master the related properties and discrimination methods of linear correlation and linear independence of vector groups.
3. Understand the concepts of maximal linearly independent group and rank of vector group, and find the maximal linearly independent group and rank of vector group.
4. Understand the concept of vector group equivalence and the relationship between the rank of matrix and the rank of its row (column) vector group.
5. Understand the concepts of N-dimensional vector space, subspace, basis, dimension and coordinate.
6. Understand the formulas of base transformation and coordinate transformation, and find the transformation matrix.
7. Understand the concept of inner product and master the Schmidt method of orthogonal normalization of linear independent vector groups.
8. Understand the concepts and properties of standard orthogonal bases and orthogonal matrices.
Fourth, linear equations.
Examination content
Cramer's Law of Linear Equations Necessary and Sufficient Conditions for Homogeneous Linear Equations to Have Non-zero Solutions Necessary and Sufficient Conditions for Non-homogeneous Linear Equations to Have Solutions Properties and Structures of Solutions of Homogeneous Linear Equations and General Solutions of Non-homogeneous Linear Equations in General Solution Space
Examination requirements
The length can be used by Kramer's law.
2. Understand the necessary and sufficient conditions for homogeneous linear equations to have nonzero solutions and nonhomogeneous linear equations to have solutions.
3. Understand the concepts of basic solution system, general solution and solution space of homogeneous linear equations, and master the solution of basic solution system and general solution of homogeneous linear equations.
4. Understand the structure of solutions of nonhomogeneous linear equations and the concept of general solutions.
5. Master the method of solving linear equations with elementary line transformation.
Eigenvalues and eigenvectors of verb (abbreviation of verb) matrix
Examination content
Concepts of eigenvalues and eigenvectors of matrices, property similarity transformation, concepts of similar matrices and necessary and sufficient conditions for similar diagonalization of property matrices, eigenvalues and eigenvectors of similar diagonal matrices and their real symmetric matrices.
Examination requirements
1. Understand the concepts and properties of eigenvalues and eigenvectors of a matrix, and you will find the eigenvalues and eigenvectors of the matrix.
2. Understand the concept and properties of similar matrix and the necessary and sufficient conditions for matrix similarity diagonalization, and master the method of transforming matrix into similar diagonal matrix.
3. Master the properties of eigenvalues and eigenvectors of real symmetric matrices.
Sixth, quadratic form
Examination content
Quadratic form and its matrix represent contract transformation and rank inertia theorem of quadratic form of contract matrix. The canonical form and canonical form of quadratic form are transformed into canonical quadratic form and the positive definiteness of its matrix by orthogonal transformation and matching method.
Examination requirements
1. Master quadratic form and its matrix representation, understand the concepts of quadratic form rank, contract transformation and contract matrix, and understand the concepts of canonical form, canonical form and inertia theorem of quadratic form.
2. Master the method of transforming quadratic form into standard form by orthogonal transformation, and can transform quadratic form into standard form by matching method.
3. Understand the concepts of positive definite quadratic form and positive definite matrix, and master their discrimination methods.
The above is the specific content of the first-line algebra examination outline for postgraduate entrance examination. I hope it will help everyone. I would like to remind you that in the final sprint stage, it is best to return to the outline, do targeted topics, and conduct more test simulations. Let's do a good job in the order and time allocation of the postgraduate examination papers. Come on!