| Section | Content |
| 1. Determinants | - Substitution elimination for linear systems, 2nd & 3rd-order determinants
- Inversion number of permutations and its properties
- Definition of general n-order determinant via inversion number
- Basic properties of n-order determinants
|
| 2. Matrix Fundamentals & Cramer's Rule | - Cofactors and algebraic cofactors
- Laplace expansion (cofactor expansion) for n-order determinants
- Calculation examples for determinants
- Cramer's Rule for n×n linear systems
- Basic concepts of matrices and vectors
|
| 3. Matrix Operations & Elementary Matrices | - Definition of matrix operations and their algebraic laws
- Verification: Associativity of matrix multiplication, transpose of product
- Block matrices
- Elementary row operations and elementary matrices
- Fundamental relationship between elementary operations and matrices
|
| 4. Matrix Rank & Invertibility | - Concept of matrix rank
- Effect of elementary operations on rank
- Invertible matrices and adjugate matrices
- Properties of invertible matrices
- Relationship between a square matrix and its adjugate
- Necessary and sufficient condition for invertibility
- Finding inverse via adjugate matrix
|
| 5. Vector Spaces I: Linear Dependence & Rank | - Elementary transformation method for inverse
- Matrix decomposition theorem
- Decomposition theorem for invertible matrices
- Rank of product with invertible matrix
- Linear combination, linear dependence/independence
- Methods to determine dependence/independence
- Relationship between linear dependence and matrix rank
- Maximal linearly independent set (basis), rank of a vector set
- Rank relations under equivalent vector sets
- Matrix rank vs. row/column vector set ranks
- Rank of matrix sum and product
|
| 6. Linear Systems I: Solvability | - Basic concepts of linear systems
- Necessary and sufficient conditions for solution existence
|
| 7. Linear Systems II: Structure & Solution | - Fundamental solution set for homogeneous systems
- Particular solution, general solution, solution structure under infinite solutions
- Solution methods
- Relationship between nonhomogeneous system solutions and solutions of its homogeneous counterpart
- Finding general solution for nonhomogeneous systems
- Matrix similarity concept
|
| 8. Eigenvalues & Eigenvectors I | - Introduction to eigenvalues and eigenvectors
- Characteristic polynomial, equation, matrix
- Methods to find eigenvalues
- Relationship between eigenvectors and fundamental solution sets
- Invariants under matrix similarity (proofs)
|
| 9. Eigenvalues & Eigenvectors II: Diagonalization | - Necessary and sufficient conditions for matrix diagonalizability
- Eigenvectors corresponding to distinct eigenvalues (linear independence)
- Maximal independent set of eigenvectors as union over eigenvalues
- Relationship between algebraic multiplicity and geometric multiplicity
|
| 10. Orthogonality & Real Symmetric Matrices | - Inner product, orthogonal vector sets
- Gram-Schmidt orthogonalization process
- Orthogonal matrices: concept and properties
- Diagonalization of real symmetric matrices
|
| 11. Quadratic Forms | - Real quadratic forms: concept, matrix representation, standard form
- Methods: Completing the square, eigenvalue method
- Congruent transformations, canonical form, inertia indices, connection to eigenvalues
|
| 12. Linear Spaces & Positive Definiteness | - Positive definite quadratic forms: concept and criteria
- Linear space: definition, basis, dimension, coordinates
- Change of basis and coordinate transformation
|
| 13. Subspaces & Linear Transformations | - Subspaces of linear spaces
- Linear transformations: definition
- Matrix representation of linear transformations
|
| 14. Eigen Theory for Linear Transformations | - Eigenvalues and eigenvectors of linear transformations
- Simplest matrix representation (Jordan form implied)
|
| 15. Inner Product Spaces | - Inner product, norm, angle between vectors
- Orthogonal sets, Euclidean spaces, orthogonal basis
- Orthogonal transformations
|
| 16. Isomorphism & Review | - Isomorphism of Euclidean spaces
- (Final Exam Review)
|
All rights reserved by 科学出版社.
