Linear Algebra Compendium¶
From freshman to PhD — a comprehensive, systematic linear algebra knowledge base
About This Site¶
This site is a comprehensive, systematic, and self-contained linear algebra knowledge base, covering all core linear algebra content from undergraduate introductory courses through doctoral-level research. Whether you are a beginner or a researcher, you can find the knowledge you need here.
The content is organized into five parts with 30+ chapters, arranged by increasing difficulty and logical coherence. Each chapter contains complete definitions, theorems, proofs, and examples, striving for rigor and clarity.
Content Guide¶
Part I: Foundations of Linear Algebra Undergraduate Basics¶
Suitable for freshmen and sophomores, covering all core content of an introductory linear algebra course.
| Chapter | Overview |
|---|---|
| Chapter 0 Polynomial Algebra | Polynomial rings, divisibility, GCD, irreducible factorization |
| Chapter 1 Systems of Linear Equations | Solving linear systems, Gaussian elimination, solution structure |
| Chapter 2 Matrices and Matrix Operations | Matrix operations, inverse matrices, block matrices, elementary matrices, rank |
| Chapter 3 Determinants | Definition and properties of determinants, cofactor expansion, Cramer's rule |
| Chapter 4 Vector Spaces | Vector space axioms, subspaces, bases and dimension, rank-nullity theorem |
| Chapter 5 Linear Transformations | Linear maps, kernel and image, matrix representation, change of basis |
| Chapter 6 Eigenvalues and Eigenvectors | Characteristic polynomial, diagonalization, Cayley–Hamilton theorem |
| Chapter 7 Orthogonality and Least Squares | Orthogonal sets, Gram–Schmidt, orthogonal projection, least squares |
Part II: Intermediate Linear Algebra Advanced Undergraduate¶
Suitable for sophomores through seniors and early graduate students, delving deeper into core linear algebra theory.
| Chapter | Overview |
|---|---|
| Chapter 8 Inner Product Spaces | General inner product spaces, orthogonal complements, adjoint operators, spectral theorem |
| Chapter 9 Quadratic and Bilinear Forms | Quadratic forms, bilinear forms, symplectic spaces, Hermitian forms |
| Chapter 10 Matrix Decompositions | LU, Cholesky, QR, Schur decompositions |
| Chapter 11 Singular Value Decomposition | SVD theory and applications, low-rank approximation, pseudoinverse |
| Chapter 12 Jordan Normal Form | Generalized eigenvectors, Jordan blocks, minimal polynomial |
| Chapter 13 Matrix Functions | Matrix exponential, matrix logarithm, matrix power series |
| Chapter 13A Quotient Spaces and Dual Spaces | Quotient spaces, dual spaces, annihilators, transpose maps, canonical isomorphism |
| Chapter 13B Lambda-Matrices and Rational Canonical Form | Lambda-matrices, Smith normal form, invariant factors, rational canonical form |
Part III: Advanced Linear Algebra Graduate¶
Suitable for master's and doctoral students, covering matrix analysis and advanced theory.
| Chapter | Overview |
|---|---|
| Chapter 14 Matrix Analysis | Matrix sequences and series, spectral radius, Gershgorin's theorem |
| Chapter 15 Norms and Perturbation Theory | Matrix norms, condition numbers, eigenvalue perturbation |
| Chapter 16 Positive Definite Matrices | Equivalent conditions for positive definiteness, Schur complement, Löwner partial order |
| Chapter 17 Nonnegative Matrices and Perron–Frobenius Theory | Perron–Frobenius theorem, irreducible matrices, stochastic matrices |
| Chapter 18 Matrix Inequalities | Eigenvalue inequalities, trace inequalities, determinantal inequalities, majorization |
| Chapter 19 Kronecker Product and Vec Operator | Kronecker product, Vec operator, and applications to matrix equations |
| Chapter 20 Matrix Equations | Sylvester equation, Lyapunov equation, Riccati equation |
Part IV: Special Topics Doctoral¶
For doctoral students and researchers, introducing frontier topics in linear algebra.
| Chapter | Overview |
|---|---|
| Chapter 21 Multilinear Algebra and Tensors | Dual spaces, tensor products, exterior algebra, tensor decomposition |
| Chapter 22 Numerical Linear Algebra | Iterative methods, Krylov subspaces, numerical stability |
| Chapter 23 Introduction to Random Matrices | Wigner semicircle law, Marchenko–Pastur law, eigenvalue distributions |
| Chapter 24 Matrix Manifolds | Stiefel manifold, Grassmann manifold, matrix Lie groups |
Part V: Applications Interdisciplinary¶
Core applications of linear algebra across disciplines.
| Chapter | Overview |
|---|---|
| Chapter 25 Linear Algebra in Optimization | Semidefinite programming, matrix completion, compressed sensing, PCA |
| Chapter 26 Linear Algebra in Differential Equations | Linear ODE systems, matrix exponential, stability analysis |
| Chapter 27 Linear Algebra in Graph Theory and Networks | Spectral graph theory, Laplacian, PageRank, expander graphs |
| Chapter 28 Linear Algebra in Quantum Information | Quantum states, unitary transformations, entanglement, quantum channels |
| Chapter 29 Linear Algebra in Statistics and Machine Learning | PCA, regression, kernel methods, dimensionality reduction |
| Chapter 30 Linear Algebra in Signal Processing and Coding | DFT, compressed sensing, error-correcting codes, wavelet transform |
Notation Conventions¶
This site uses the following standard mathematical notation:
| Symbol | Meaning |
|---|---|
| \(\mathbb{R}, \mathbb{C}, \mathbb{F}\) | Real numbers, complex numbers, general field |
| \(\mathbb{R}^n, \mathbb{R}^{m \times n}\) | \(n\)-dimensional real vector space, \(m \times n\) real matrix space |
| \(A, B, C\) | Matrices (uppercase letters) |
| \(\mathbf{v}, \mathbf{u}, \mathbf{w}\) | Vectors (boldface lowercase letters) |
| \(a, b, \lambda, \alpha\) | Scalars (lowercase or Greek letters) |
| \(V, W, U\) | Vector spaces |
| \(T, S\) | Linear transformations |
| \(A^T, A^H\) | Transpose, conjugate transpose (Hermitian transpose) |
| \(A^{-1}\) | Inverse matrix |
| \(\det(A)\) | Determinant |
| \(\operatorname{tr}(A)\) | Trace |
| \(\operatorname{rank}(A)\) | Rank |
| \(\dim(V)\) | Dimension |
| \(\ker(T), \operatorname{im}(T)\) | Kernel (null space), image (range) |
| \(\langle \mathbf{u}, \mathbf{v} \rangle\) | Inner product |
| \(\|\mathbf{v}\|\) | Norm |
| \(\sigma_i(A)\) | The \(i\)-th singular value |
| \(\lambda_i(A)\) | The \(i\)-th eigenvalue |
| \(I_n\) or \(I\) | \(n \times n\) identity matrix |
| \(O\) | Zero matrix |
| \(\mathbf{0}\) | Zero vector |
| \(\oplus\) | Direct sum |
| \(\otimes\) | Tensor product / Kronecker product |
How to Use This Site¶
- Search: Use the search bar at the top of the page to quickly find any concept, theorem, or keyword. Supports both Chinese and English search.
- Navigation: Browse by chapter using the left sidebar, or use the right-hand table of contents to jump to a specific section on the current page.
- Light/Dark Mode: Click the brightness icon at the top to switch between light and dark modes.
- Math Formulas: All formulas on this site are rendered with MathJax. You can copy the LaTeX source by right-clicking a formula.
References¶
The content of this site draws from the following classic textbooks:
- Strang, G. Introduction to Linear Algebra. Wellesley-Cambridge Press.
- Lay, D.C. Linear Algebra and Its Applications. Pearson.
- Axler, S. Linear Algebra Done Right. Springer.
- Hoffman, K. & Kunze, R. Linear Algebra. Prentice Hall.
- Horn, R.A. & Johnson, C.R. Matrix Analysis. Cambridge University Press.
- Horn, R.A. & Johnson, C.R. Topics in Matrix Analysis. Cambridge University Press.
- Meyer, C.D. Matrix Analysis and Applied Linear Algebra. SIAM.
- Golub, G.H. & Van Loan, C.F. Matrix Computations. Johns Hopkins University Press.
- Halmos, P.R. Finite-Dimensional Vector Spaces. Springer.
- Lax, P.D. Linear Algebra and Its Applications. Wiley.