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Linear Algebra

Vectors, matrices, eigenvalues, and the geometry that ties them together — from Gaussian elimination through spectral theory to PCA, graphs, and differential equations.

intermediate Free mathlinear-algebra
📖 51 readings ⚡ 51 exercises · 500 min total
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9 modules · 51 lessons
16. Foundations 6 lessons · 66 min
  1. 1 Vectors in ℝⁿ: operations and geometric interpretation 12 min
  2. 2 Systems of linear equations, Gaussian elimination, row-echelon form (RREF) 12 min
  3. 3 Matrices and matrix operations (addition, multiplication, transpose) 11 min
  4. 4 Matrix inverse, invertibility conditions 10 min
  5. 5 Determinants: definition, cofactor expansion, properties, geometric meaning, Cramer's Rule 10 min
  6. 6 Elementary matrices and LU decomposition 11 min
17. Vector Spaces 6 lessons · 60 min
  1. 7 Vector spaces and subspaces (abstract definition, not just ℝⁿ) 10 min
  2. 8 Span, linear independence 10 min
  3. 9 Basis and dimension 11 min
  4. 10 Coordinate vectors relative to a basis, change of basis 9 min
  5. 11 Null space, column space, row space 10 min
  6. 12 Rank-Nullity Theorem 10 min
18. Linear Transformations 6 lessons · 57 min
  1. 13 Definition and examples; linearity conditions 10 min
  2. 14 Matrix representation of a linear transformation 9 min
  3. 15 Geometric transformations: rotation, reflection, scaling, shear 10 min
  4. 16 Kernel and image 9 min
  5. 17 Isomorphisms, invertible linear maps 9 min
  6. 18 Composition of transformations 10 min
19. Eigenvalues and Eigenvectors 6 lessons · 55 min
  1. 19 Characteristic polynomial, eigenvalues, eigenvectors 9 min
  2. 20 Diagonalization, when a matrix is diagonalizable 9 min
  3. 21 Eigenspaces; algebraic vs. geometric multiplicity 9 min
  4. 22 Similar matrices and invariants 10 min
  5. 23 Adv: generalized eigenvectors and Jordan canonical form 9 min
  6. 24 Adv: Cayley-Hamilton Theorem 9 min
20. Inner Product Spaces 7 lessons · 66 min
  1. 25 Dot product, norms, angles between vectors 9 min
  2. 26 General inner products (abstract inner product spaces) 10 min
  3. 27 Orthogonality, orthogonal complements 9 min
  4. 28 Gram-Schmidt orthogonalization process 10 min
  5. 29 Orthogonal and orthonormal bases 9 min
  6. 30 Least squares approximation and orthogonal projections 10 min
  7. 31 QR decomposition 9 min
21. Spectral Theory and Special Matrices 6 lessons · 55 min
  1. 32 Symmetric matrices and the Spectral Theorem 9 min
  2. 33 Adv: Hermitian and unitary matrices (complex case) 9 min
  3. 34 Positive definite / semidefinite matrices 10 min
  4. 35 Quadratic forms, classification via eigenvalues 9 min
  5. 36 Singular Value Decomposition (SVD) 9 min
  6. 37 Adv: normal matrices, spectral theorem in full generality 9 min
22. Matrix Decompositions & Numerical Linear Algebra (Adv) 5 lessons · 49 min
  1. 38 LU, Cholesky, QR decompositions (unified review) 10 min
  2. 39 Schur decomposition 9 min
  3. 40 Matrix norms and condition numbers 10 min
  4. 41 Iterative methods (intro): power iteration for dominant eigenvalue 10 min
  5. 42 Adv: Perron-Frobenius Theorem (intro, for nonnegative matrices) 10 min
23. Bilinear/Multilinear Structures (Adv) 4 lessons · 42 min
  1. 43 Dual spaces and dual bases 10 min
  2. 44 Bilinear forms 10 min
  3. 45 Adv: tensor products, intro to multilinear algebra 10 min
  4. 46 Adv: linear algebra over general fields (bridge to abstract algebra) 12 min
24. Applications 5 lessons · 50 min
  1. 47 Markov chains and stochastic matrices (bridge to probability course) 10 min
  2. 48 Principal Component Analysis (PCA) via SVD/eigen-decomposition 10 min
  3. 49 Graph theory: adjacency and Laplacian matrices, spectral graph theory (intro) 10 min
  4. 50 Linear regression as a least-squares linear algebra problem (bridge to stats course) 10 min
  5. 51 Systems of linear ODEs via eigen-decomposition (bridge to calculus course) 10 min

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