Second, the main contents of the review:
1. Determinants
Definition, properties and common calculation methods of determinant (such as triangulation, edge addition, order reduction, recursion, split term, Vandermonde determinant, mathematical induction, auxiliary determinant).
Emphasis: the calculation of n-order determinant.
2. Matrix theory
Matrix operation, elementary transformation of block matrix and rank of matrix, invertible matrix and adjoint matrix, three equivalence relations of matrix (equivalence, contraction and similarity), eigenvalue and eigenvector of matrix, trace of matrix, minimum polynomial of matrix, diagonalization of matrix, common decomposition of matrix (such as equivalent decomposition, full rank decomposition, orthogonal similarity decomposition of real symmetric matrix, orthogonal triangle decomposition of real invertible matrix, Jordan decomposition).
Emphasis: using the elementary transformation of block matrix to prove the equality and inequality of matrix rank, the properties and solutions of matrix inverse matrix and adjoint matrix, the relationship of three equivalent relationships of matrix, the judgment and proof of matrix diagonalization (especially the simultaneous diagonalization of multiple matrices), the proof and application of matrix decomposition (especially the orthogonal similar decomposition of real symmetric matrix, the calculation and related proof of Jordan standard form).
3. Linear equation
Cramer's rule, necessary and sufficient conditions for homogeneous linear equations to have non-zero solutions, solutions of basic solution systems and related proofs, solutions of nonhomogeneous linear equations and the structure of solutions.
Emphasis: the structure of solutions of nonhomogeneous linear equations and the proof of the basic solution system of their derivative groups. Solutions of special equations.
4. Polynomial theory
Divisibility of polynomials, greatest common factor and least common multiple, coprime of polynomials, irreducible polynomials and factorization, roots of polynomial functions and polynomials.
Emphasis: using polynomial theory to prove related problems, such as the proof and application of coprime and irreducible polynomial properties; Proof of important theorems, such as uniqueness theorem of factorization, eisenstein discriminant, Gauss lemma, proof of irreducible polynomials.
5. Quadratic theory
Quadratic linear space and symmetric matrix space are isomorphic, quadratic form is standard form and standard form, Sylvester's law of inertia, definitions and properties of positive definite, semi-positive definite, negative definite, semi-negative definite and indefinite quadratic form, some important conclusions of positive definite matrix and their applications.
Emphasis: the proof of positive definite and semi-positive definite matrices, the classification of N-level square matrices according to contractual relations, and the proof of real symmetric matrices.
6. Linear space and Euclidean space
Definition of linear space, linear relation of vector group (linear correlation and linear independence, equivalence of vector group, solution of maximum linear independence group, substitution theorem), basis and extension basis theorem, dimension formula, coordinate transformation, basis transformation and coordinate transformation, generating subspace, intersection and sum of subspace (including direct sum), definition and simple properties of inner product and Euclidean space, orthogonal complement of subspace, solution of metric matrix, and so on.
Emphasis: comprehensive proof of linear correlation and linear independence of vector groups, judging whether a vector is represented by a group of vectors and how to represent it, finding the largest independent group of vector groups and using it to represent other vectors, proof and application of dimension formula, especially proof of direct sum of subspaces, solution of standard orthogonal basis and proof of its properties.
7. Linear transformation
Definition, operation and matrix of linear transformation, kernel and range of linear transformation, invariant subspace, characteristic root and vector of linear transformation, characteristic subspace, diagonalization of linear transformation, orthogonal transformation, symmetric transformation and antisymmetric transformation, application of corresponding relationship between linear transformation and its matrix, and its eigenvalue and eigenvector.
Emphasis: the application of the corresponding relationship between linear transformation and its matrix, the diagonalization of linear transformation, the core and scope of linear transformation.
Proof of orthogonal transformation, symmetric transformation and antisymmetric transformation. The relationship between minimum polynomial and diagonalization.