[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"skill:01M6E53GK2BDZCMS7ENM65GD66":3,"skill-same:01M6E53GK2BDZCMS7ENM65GD66":80},{"skillNo":4,"name":5,"nameZh":6,"description":7,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":16,"categories":17,"keywords":18,"publishedAt":39,"linkOnly":40,"markdownContent":41,"markdownBody":42,"frontMatter":43,"refName":48,"commitSha":49,"skillDirectory":50,"forksCount":51,"repositoryHtmlUrl":52,"repositoryHomepageUrl":6,"ownerHtmlUrl":53,"ownerBlogUrl":6,"ownerDescription":54,"updatedAt":55,"install":56,"resources":69},"01M6E53GK2BDZCMS7ENM65GD66","linear-algebra-expert",null,"Expert-level linear algebra knowledge. Use when working with vectors, matrices, determinants, eigenvalues, linear transformations, vector spaces, inner products, decompositions, or applications in machine learning and physics. Also use when the user mentions 'matrix', 'determinant', 'eigenvalue', 'eigenvector', 'linear transformation', 'vector space', 'basis', 'orthogonality', 'SVD', 'PCA', 'least squares', 'null space', or 'rank'.","luokai0\u002Fai-agent-skills-by-luo-kai","luokai0","https:\u002F\u002Favatars.githubusercontent.com\u002Fu\u002F266954973?v=4","01M6E16VNT42JB8B3X3254QWJ7","luo kai","01M6E16VTFVP6S5XJF1V1MPV90",14,"MIT","https:\u002F\u002Fgithub.com\u002Fluokai0\u002Fai-agent-skills-by-luo-kai\u002Fblob\u002Fmain\u002Fai-agent-skills\u002F13-physics-and-mathematics (by Luo Kai)\u002Flinear-algebra-expert\u002FSKILL.md",[],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],"linear","linear algebra expert","linear algebra","algebra","expert","learning","knowledge","user","use","science","physics","applications","vector","matrices","vectors","eigenvalues","transformation","PCA","machine","machine learning","2026-11-10 10:12:30",false,"---\nauthor: luo-kai\nname: linear-algebra-expert\ndescription: Expert-level linear algebra knowledge. Use when working with vectors, matrices, determinants, eigenvalues, linear transformations, vector spaces, inner products, decompositions, or applications in machine learning and physics. Also use when the user mentions 'matrix', 'determinant', 'eigenvalue', 'eigenvector', 'linear transformation', 'vector space', 'basis', 'orthogonality', 'SVD', 'PCA', 'least squares', 'null space', or 'rank'.\nlicense: MIT\nmetadata:\n  author: luokai25\n  version: \"1.0\"\n  category: science\n---\n\n# Linear Algebra Expert\n\nYou are a world-class mathematician with deep expertise in linear algebra covering vectors, matrices, linear transformations, eigentheory, decompositions, inner product spaces, and applications in data science, physics, and engineering.\n\n## Before Starting\n\n1. **Topic** — Vectors, matrices, eigenvalues, decompositions, or applications?\n2. **Level** — High school, undergraduate, or graduate?\n3. **Goal** — Solve system, understand concept, or apply to ML\u002Fphysics?\n4. **Context** — Pure math, data science, physics, or engineering?\n5. **Computation** — Theoretical or numerical\u002Fcomputational?\n\n---\n\n## Core Expertise Areas\n\n- **Vectors & Matrices**: operations, properties, special matrices\n- **Systems of Equations**: Gaussian elimination, row reduction, solutions\n- **Determinants**: computation, properties, geometric meaning\n- **Vector Spaces**: subspaces, basis, dimension, rank\n- **Linear Transformations**: kernel, image, matrix representation\n- **Eigentheory**: eigenvalues, eigenvectors, diagonalization\n- **Inner Product Spaces**: dot product, orthogonality, Gram-Schmidt\n- **Matrix Decompositions**: LU, QR, SVD, spectral theorem\n\n---\n\n## Vectors & Vector Operations\n```\nVector: ordered list of numbers v = (v₁, v₂, ..., vₙ) ∈ ℝⁿ\n\nOperations:\n  Addition:      u + v = (u₁+v₁, u₂+v₂, ..., uₙ+vₙ)\n  Scalar mult:   cv = (cv₁, cv₂, ..., cvₙ)\n  Dot product:   u·v = Σuᵢvᵢ = |u||v|cosθ\n  Cross product: u×v = (u₂v₃-u₃v₂, u₃v₁-u₁v₃, u₁v₂-u₂v₁)  (ℝ³ only)\n  Norm:          |v| = √(v·v) = √(Σvᵢ²)\n\nProperties of dot product:\n  Commutative:  u·v = v·u\n  Distributive: u·(v+w) = u·v + u·w\n  u·v = 0 ↔ u ⊥ v  (orthogonal)\n  Cauchy-Schwarz: |u·v| ≤ |u||v|\n  Triangle inequality: |u+v| ≤ |u| + |v|\n\nProjection:\n  proj_v u = (u·v\u002Fv·v) v  (projection of u onto v)\n  comp_v u = u·v\u002F|v|  (scalar component)\n\nLinear independence:\n  {v₁,...,vₖ} linearly independent if:\n  c₁v₁ + c₂v₂ + ... + cₖvₖ = 0 → all cᵢ = 0\n  Dependent: one vector is linear combination of others\n```\n\n---\n\n## Matrix Operations\n```\nMatrix A is m×n (m rows, n columns)\n  Entry: aᵢⱼ (row i, column j)\n\nOperations:\n  Addition:      (A+B)ᵢⱼ = aᵢⱼ + bᵢⱼ  (same size)\n  Scalar mult:   (cA)ᵢⱼ = c·aᵢⱼ\n  Transpose:     (Aᵀ)ᵢⱼ = aⱼᵢ\n  Multiplication: (AB)ᵢⱼ = Σₖ aᵢₖbₖⱼ  (A is m×p, B is p×n)\n\nMatrix multiplication properties:\n  NOT commutative: AB ≠ BA in general\n  Associative:     (AB)C = A(BC)\n  Distributive:    A(B+C) = AB + AC\n  Transpose:       (AB)ᵀ = BᵀAᵀ\n  Inverse:         (AB)⁻¹ = B⁻¹A⁻¹\n\nSpecial matrices:\n  Zero matrix: O (all zeros)\n  Identity: I (1s on diagonal)\n  Diagonal: D (nonzero only on diagonal)\n  Symmetric: A = Aᵀ\n  Skew-symmetric: A = -Aᵀ\n  Orthogonal: AᵀA = I (columns are orthonormal)\n  Unitary: A*A = I (complex: conjugate transpose)\n\nBlock matrices:\n  Partition matrices into blocks for computation\n  Block diagonal: [[A 0][0 B]]\n  Block multiplication follows matrix multiplication rules\n```\n\n---\n\n## Systems of Linear Equations\n```\nSystem: Ax = b\n  A: m×n coefficient matrix\n  x: n×1 unknown vector\n  b: m×1 right-hand side\n\nAugmented matrix: [A|b]\n\nRow operations (preserve solution set):\n  R1: Multiply row by nonzero scalar\n  R2: Add multiple of one row to another\n  R3: Swap two rows\n\nRow Echelon Form (REF):\n  Leading nonzero in each row to right of row above\n  Zero rows at bottom\n\nReduced Row Echelon Form (RREF):\n  Each pivot = 1\n  Each pivot is only nonzero in its column\n\nSolution types:\n  Unique solution:   rref gives [I|c], n pivots\n  Infinitely many:   free variables (fewer pivots than unknowns)\n  No solution:       inconsistent row [0 0...0 | b≠0]\n\nRank-Nullity theorem:\n  rank(A) + nullity(A) = n  (number of columns)\n  rank = number of pivots\n  nullity = dimension of null space (number of free variables)\n\nCramer's rule (n×n, det(A)≠0):\n  xᵢ = det(Aᵢ)\u002Fdet(A)  (Aᵢ = A with column i replaced by b)\n  Impractical for large n (use Gaussian elimination)\n```\n\n---\n\n## Determinants\n```\n2×2: det[[a b][c d]] = ad - bc\n\n3×3 (cofactor expansion along row 1):\n  det(A) = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃\n  Cᵢⱼ = (-1)^(i+j) Mᵢⱼ  (cofactor, Mᵢⱼ = minor)\n\nProperties:\n  det(Aᵀ) = det(A)\n  det(AB) = det(A)det(B)\n  det(A⁻¹) = 1\u002Fdet(A)\n  det(cA) = cⁿdet(A)  (n×n matrix)\n  Row swap: det changes sign\n  Row multiply by k: det multiplied by k\n  Add multiple of row to another: det unchanged\n  Triangular matrix: det = product of diagonal entries\n\nGeometric meaning:\n  2×2: |det(A)| = area of parallelogram spanned by rows\u002Fcolumns\n  3×3: |det(A)| = volume of parallelepiped\n  det(A) > 0: orientation preserved\n  det(A) \u003C 0: orientation reversed\n  det(A) = 0: matrix is singular (rows\u002Fcolumns linearly dependent)\n\nInverse formula:\n  A⁻¹ = (1\u002Fdet(A)) · adj(A)\n  adj(A) = (Cᵢⱼ)ᵀ  (adjugate = transpose of cofactor matrix)\n```\n\n---\n\n## Vector Spaces\n```\nVector space V over field F:\n  Satisfies 8 axioms (closure, commutativity, associativity,\n  identity, inverse for + ; associativity, distributivity, identity for ·)\n\nExamples:\n  ℝⁿ: n-tuples of real numbers\n  Mₘₙ: m×n matrices\n  Pₙ: polynomials of degree ≤ n\n  C[a,b]: continuous functions on [a,b]\n  Solution set of Ax = 0 (null space)\n\nSubspace: subset W of V satisfying:\n  0 ∈ W\n  u,v ∈ W → u+v ∈ W\n  u ∈ W → cu ∈ W  (for all scalars c)\n\nFour fundamental subspaces of A (m×n):\n  Column space C(A): span of columns, subspace of ℝᵐ\n  Row space R(A): span of rows, subspace of ℝⁿ\n  Null space N(A): {x: Ax=0}, subspace of ℝⁿ\n  Left null space N(Aᵀ): {y: Aᵀy=0}, subspace of ℝᵐ\n\n  dim(C(A)) = dim(R(A)) = rank(A)  (fundamental theorem)\n  N(A) ⊥ R(A) and N(Aᵀ) ⊥ C(A)\n\nBasis and dimension:\n  Basis: linearly independent spanning set\n  Dimension: number of vectors in any basis\n  Standard basis ℝⁿ: e₁=(1,0,...,0), ..., eₙ=(0,...,0,1)\n  Change of basis: P·[x]_B = [x]_standard\n```\n\n---\n\n## Eigentheory\n```\nEigenvalue equation: Av = λv\n  λ = eigenvalue (scalar)\n  v = eigenvector (nonzero vector)\n  (A - λI)v = 0 → det(A - λI) = 0\n\nCharacteristic polynomial:\n  p(λ) = det(A - λI) = 0\n  Degree n for n×n matrix → n eigenvalues (counting multiplicity)\n  Roots may be complex even for real A\n\nFinding eigenvectors:\n  For each eigenvalue λᵢ: solve (A-λᵢI)v = 0\n  Eigenspace = null space of (A-λᵢI)\n\nProperties:\n  Trace(A) = sum of eigenvalues = Σλᵢ\n  det(A) = product of eigenvalues = Πλᵢ\n  Similar matrices have same eigenvalues\n  Symmetric A: real eigenvalues, orthogonal eigenvectors\n  Symmetric A: positive definite ↔ all eigenvalues > 0\n\nDiagonalization:\n  A = PDP⁻¹  where D = diag(λ₁,...,λₙ)\n  P = matrix of eigenvectors as columns\n  Requires n linearly independent eigenvectors\n  If A has n distinct eigenvalues → diagonalizable\n\nPowers and functions of matrices:\n  Aᵏ = PDᵏP⁻¹  (Dᵏ = diag(λ₁ᵏ,...,λₙᵏ))\n  eᴬ = PeᴰP⁻¹  (eᴰ = diag(eˡ¹,...,eˡⁿ))\n\nSpectral theorem:\n  A symmetric → A = QΛQᵀ  (Q orthogonal, Λ diagonal)\n  Eigenvectors orthonormal, eigenvalues real\n```\n\n---\n\n## Inner Product Spaces\n```\nInner product ⟨u,v⟩:\n  Conjugate symmetry: ⟨u,v⟩ = ⟨v,u⟩* (or symmetric for real)\n  Linearity: ⟨au+bw,v⟩ = a⟨u,v⟩ + b⟨w,v⟩\n  Positive definite: ⟨v,v⟩ ≥ 0, = 0 ↔ v = 0\n\nStandard inner product (dot product): ⟨u,v⟩ = u·v = Σuᵢvᵢ\nFunction inner product: ⟨f,g⟩ = ∫ₐᵇ f(x)g(x)dx\n\nNorm: ||v|| = √⟨v,v⟩\nCauchy-Schwarz: |⟨u,v⟩| ≤ ||u|| ||v||\nAngle: cosθ = ⟨u,v⟩\u002F(||u|| ||v||)\n\nOrthogonality:\n  u ⊥ v ↔ ⟨u,v⟩ = 0\n  Orthogonal set: mutually orthogonal\n  Orthonormal set: orthogonal + each ||vᵢ|| = 1\n  Pythagorean theorem: ||u+v||² = ||u||² + ||v||²  if u⊥v\n\nGram-Schmidt process:\n  Input: {v₁,...,vₙ} linearly independent\n  Output: {u₁,...,uₙ} orthonormal basis\n\n  u₁ = v₁\u002F||v₁||\n  w₂ = v₂ - ⟨v₂,u₁⟩u₁\n  u₂ = w₂\u002F||w₂||\n  wₖ = vₖ - Σᵢ₌₁^(k-1) ⟨vₖ,uᵢ⟩uᵢ\n  uₖ = wₖ\u002F||wₖ||\n\nOrthogonal complement:\n  W⊥ = {v: ⟨v,w⟩=0 ∀w∈W}\n  V = W ⊕ W⊥ (direct sum)\n  Projection: proj_W v = Σᵢ⟨v,uᵢ⟩uᵢ  (orthonormal basis {uᵢ} for W)\n```\n\n---\n\n## Matrix Decompositions\n```python\ndef matrix_decompositions():\n    return {\n        'LU Decomposition': {\n            'form':         'A = LU  (L lower triangular, U upper triangular)',\n            'use':          'Solve Ax=b: Ly=b, then Ux=y',\n            'advantage':    'Reuse factorization for multiple b vectors',\n            'with_pivoting':'PA = LU  (P = permutation, for numerical stability)',\n            'complexity':   'O(n³) once, O(n²) per solve'\n        },\n        'QR Decomposition': {\n            'form':         'A = QR  (Q orthogonal, R upper triangular)',\n            'method':       'Gram-Schmidt, Householder, Givens rotations',\n            'use':          'Least squares, eigenvalue algorithms',\n            'stability':    'More numerically stable than LU for least squares'\n        },\n        'Eigendecomposition': {\n            'form':         'A = PDP⁻¹  (only for diagonalizable A)',\n            'symmetric':    'A = QΛQᵀ  (spectral theorem, Q orthogonal)',\n            'use':          'Matrix powers, differential equations, PCA'\n        },\n        'SVD (Singular Value Decomposition)': {\n            'form':         'A = UΣVᵀ  (any m×n matrix)',\n            'U':            'm×m orthogonal (left singular vectors)',\n            'Σ':            'm×n diagonal (singular values σ₁≥σ₂≥...≥0)',\n            'V':            'n×n orthogonal (right singular vectors)',\n            'rank':         'rank(A) = number of nonzero singular values',\n            'uses': [\n                'Low-rank approximation: Aₖ = UₖΣₖVₖᵀ (best rank-k approximation)',\n                'PCA: SVD of centered data matrix',\n                'Pseudoinverse: A⁺ = VΣ⁺Uᵀ',\n                'Least squares: x = A⁺b',\n                'Image compression, recommender systems, NLP (LSA)'\n            ]\n        },\n        'Cholesky': {\n            'form':         'A = LLᵀ  (A symmetric positive definite)',\n            'advantage':    'Twice as fast as LU, numerically stable',\n            'use':          'SPD systems, multivariate statistics'\n        },\n        'Jordan Normal Form': {\n            'form':         'A = PJP⁻¹  (J = Jordan blocks)',\n            'use':          'Non-diagonalizable matrices, theoretical analysis',\n            'jordan_block':  'Jλ = [[λ 1 0][0 λ 1][0 0 λ]] (λ on diagonal, 1 above)'\n        }\n    }\n```\n\n---\n\n## Least Squares & Applications\n```\nLeast squares problem:\n  Ax = b has no solution (overdetermined, m > n)\n  Find x̂ minimizing ||Ax - b||²\n  Normal equations: AᵀAx̂ = Aᵀb\n  Solution: x̂ = (AᵀA)⁻¹Aᵀb  (if AᵀA invertible)\n  Via SVD: x̂ = A⁺b = VΣ⁺Uᵀb\n\nPCA (Principal Component Analysis):\n  Center data: subtract mean\n  Compute covariance matrix C = (1\u002Fn)XᵀX\n  Eigendecompose: C = QΛQᵀ\n  Or SVD of X: X = UΣVᵀ\n  Principal components = columns of V (or Q)\n  Variance explained by kth PC = σₖ²\u002FΣσᵢ²\n\nLinear transformations:\n  T: ℝⁿ → ℝᵐ is linear if T(u+v)=T(u)+T(v), T(cu)=cT(u)\n  Every linear transformation: T(x) = Ax for some matrix A\n  Kernel (null space): ker(T) = {x: T(x)=0}\n  Image (range): im(T) = {T(x): x∈ℝⁿ} = C(A)\n  Rank-Nullity: dim(ker) + dim(im) = n\n\nQuadratic forms:\n  Q(x) = xᵀAx  (A symmetric)\n  Positive definite: Q(x) > 0 ∀x≠0 ↔ all eigenvalues > 0\n  Negative definite: all eigenvalues \u003C 0\n  Indefinite: mixed eigenvalue signs\n  Classification by eigenvalues of A\n```\n\n---\n\n## Numerical Linear Algebra\n```python\ndef numerical_methods_linalg():\n    return {\n        'Condition number': {\n            'definition':   'κ(A) = ||A|| · ||A⁻¹|| = σ_max\u002Fσ_min',\n            'interpretation': 'How sensitive solution is to perturbations in b',\n            'ill-conditioned': 'κ >> 1: small error in b → large error in x',\n            'well-conditioned': 'κ ≈ 1: stable computation'\n        },\n        'Iterative methods (large sparse A)': {\n            'Jacobi':       'Diagonal update, simple but slow',\n            'Gauss-Seidel': 'Use updated values immediately, faster',\n            'CG (Conjugate Gradient)': 'For SPD matrices, optimal convergence',\n            'GMRES':        'For non-symmetric, Krylov subspace method',\n            'Convergence':  'Depends on eigenvalue distribution and condition number'\n        },\n        'Power iteration (eigenvalues)': {\n            'method':       'Repeatedly multiply by A and normalize',\n            'finds':        'Largest eigenvalue and corresponding eigenvector',\n            'QR algorithm': 'Standard for all eigenvalues, O(n³)',\n            'Lanczos':      'For large sparse symmetric matrices'\n        },\n        'Floating point issues': {\n            'loss_of_significance': 'Subtracting nearly equal numbers',\n            'overflow_underflow':   'Extremely large\u002Fsmall intermediate values',\n            'pivoting':             'Partial pivoting prevents division by small numbers'\n        }\n    }\n```\n\n---\n\n## Key Theorems Summary\n```\nInvertible Matrix Theorem (all equivalent for n×n A):\n  A is invertible ↔\n  det(A) ≠ 0 ↔\n  Ax = b has unique solution for every b ↔\n  Ax = 0 has only trivial solution ↔\n  rank(A) = n ↔\n  columns of A are linearly independent ↔\n  rows of A are linearly independent ↔\n  0 is not an eigenvalue of A ↔\n  AᵀA is invertible ↔\n  A = product of elementary matrices\n\nRank-Nullity: rank(A) + nullity(A) = n  (# columns)\nDimension theorem: dim(C(A)) = dim(R(A)) = rank(A)\nSpectral theorem: Symmetric A → orthogonally diagonalizable\nSVD: Every A = UΣVᵀ (exists for any matrix)\n```\n\n---\n\n## Common Pitfalls\n\n| Pitfall | Fix |\n|---|---|\n| AB = BA assumed | Matrix multiplication is NOT commutative |\n| (A+B)² = A²+2AB+B² | Only if AB = BA; general: (A+B)² = A²+AB+BA+B² |\n| det(A+B) = det(A)+det(B) | WRONG: determinant is NOT linear in this way |\n| Eigenvalues of AB = product of eigenvalues | Only true if A and B share eigenvectors |\n| Rank = number of nonzero rows | Rank = number of nonzero rows in REF |\n| Orthogonal = perpendicular only | Orthogonal matrix has columns forming orthonormal set |\n\n---\n\n## Related Skills\n\n- **calculus-expert**: Multivariable and vector calculus\n- **statistics-expert**: PCA, regression, covariance matrices\n- **quantum-mechanics-expert**: Linear operators in Hilbert space\n- **machine-learning-expert**: SVD, PCA, neural networks\n- **numerical-methods-expert**: Numerical linear algebra\n- **differential-equations-expert**: Systems of ODEs using matrices\n","# Linear Algebra Expert\n\nYou are a world-class mathematician with deep expertise in linear algebra covering vectors, matrices, linear transformations, eigentheory, decompositions, inner product spaces, and applications in data science, physics, and engineering.\n\n## Before Starting\n\n1. **Topic** — Vectors, matrices, eigenvalues, decompositions, or applications?\n2. **Level** — High school, undergraduate, or graduate?\n3. **Goal** — Solve system, understand concept, or apply to ML\u002Fphysics?\n4. **Context** — Pure math, data science, physics, or engineering?\n5. **Computation** — Theoretical or numerical\u002Fcomputational?\n\n---\n\n## Core Expertise Areas\n\n- **Vectors & Matrices**: operations, properties, special matrices\n- **Systems of Equations**: Gaussian elimination, row reduction, solutions\n- **Determinants**: computation, properties, geometric meaning\n- **Vector Spaces**: subspaces, basis, dimension, rank\n- **Linear Transformations**: kernel, image, matrix representation\n- **Eigentheory**: eigenvalues, eigenvectors, diagonalization\n- **Inner Product Spaces**: dot product, orthogonality, Gram-Schmidt\n- **Matrix Decompositions**: LU, QR, SVD, spectral theorem\n\n---\n\n## Vectors & Vector Operations\n```\nVector: ordered list of numbers v = (v₁, v₂, ..., vₙ) ∈ ℝⁿ\n\nOperations:\n  Addition:      u + v = (u₁+v₁, u₂+v₂, ..., uₙ+vₙ)\n  Scalar mult:   cv = (cv₁, cv₂, ..., cvₙ)\n  Dot product:   u·v = Σuᵢvᵢ = |u||v|cosθ\n  Cross product: u×v = (u₂v₃-u₃v₂, u₃v₁-u₁v₃, u₁v₂-u₂v₁)  (ℝ³ only)\n  Norm:          |v| = √(v·v) = √(Σvᵢ²)\n\nProperties of dot product:\n  Commutative:  u·v = v·u\n  Distributive: u·(v+w) = u·v + u·w\n  u·v = 0 ↔ u ⊥ v  (orthogonal)\n  Cauchy-Schwarz: |u·v| ≤ |u||v|\n  Triangle inequality: |u+v| ≤ |u| + |v|\n\nProjection:\n  proj_v u = (u·v\u002Fv·v) v  (projection of u onto v)\n  comp_v u = u·v\u002F|v|  (scalar component)\n\nLinear independence:\n  {v₁,...,vₖ} linearly independent if:\n  c₁v₁ + c₂v₂ + ... + cₖvₖ = 0 → all cᵢ = 0\n  Dependent: one vector is linear combination of others\n```\n\n---\n\n## Matrix Operations\n```\nMatrix A is m×n (m rows, n columns)\n  Entry: aᵢⱼ (row i, column j)\n\nOperations:\n  Addition:      (A+B)ᵢⱼ = aᵢⱼ + bᵢⱼ  (same size)\n  Scalar mult:   (cA)ᵢⱼ = c·aᵢⱼ\n  Transpose:     (Aᵀ)ᵢⱼ = aⱼᵢ\n  Multiplication: (AB)ᵢⱼ = Σₖ aᵢₖbₖⱼ  (A is m×p, B is p×n)\n\nMatrix multiplication properties:\n  NOT commutative: AB ≠ BA in general\n  Associative:     (AB)C = A(BC)\n  Distributive:    A(B+C) = AB + AC\n  Transpose:       (AB)ᵀ = BᵀAᵀ\n  Inverse:         (AB)⁻¹ = B⁻¹A⁻¹\n\nSpecial matrices:\n  Zero matrix: O (all zeros)\n  Identity: I (1s on diagonal)\n  Diagonal: D (nonzero only on diagonal)\n  Symmetric: A = Aᵀ\n  Skew-symmetric: A = -Aᵀ\n  Orthogonal: AᵀA = I (columns are orthonormal)\n  Unitary: A*A = I (complex: conjugate transpose)\n\nBlock matrices:\n  Partition matrices into blocks for computation\n  Block diagonal: [[A 0][0 B]]\n  Block multiplication follows matrix multiplication rules\n```\n\n---\n\n## Systems of Linear Equations\n```\nSystem: Ax = b\n  A: m×n coefficient matrix\n  x: n×1 unknown vector\n  b: m×1 right-hand side\n\nAugmented matrix: [A|b]\n\nRow operations (preserve solution set):\n  R1: Multiply row by nonzero scalar\n  R2: Add multiple of one row to another\n  R3: Swap two rows\n\nRow Echelon Form (REF):\n  Leading nonzero in each row to right of row above\n  Zero rows at bottom\n\nReduced Row Echelon Form (RREF):\n  Each pivot = 1\n  Each pivot is only nonzero in its column\n\nSolution types:\n  Unique solution:   rref gives [I|c], n pivots\n  Infinitely many:   free variables (fewer pivots than unknowns)\n  No solution:       inconsistent row [0 0...0 | b≠0]\n\nRank-Nullity theorem:\n  rank(A) + nullity(A) = n  (number of columns)\n  rank = number of pivots\n  nullity = dimension of null space (number of free variables)\n\nCramer's rule (n×n, det(A)≠0):\n  xᵢ = det(Aᵢ)\u002Fdet(A)  (Aᵢ = A with column i replaced by b)\n  Impractical for large n (use Gaussian elimination)\n```\n\n---\n\n## Determinants\n```\n2×2: det[[a b][c d]] = ad - bc\n\n3×3 (cofactor expansion along row 1):\n  det(A) = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃\n  Cᵢⱼ = (-1)^(i+j) Mᵢⱼ  (cofactor, Mᵢⱼ = minor)\n\nProperties:\n  det(Aᵀ) = det(A)\n  det(AB) = det(A)det(B)\n  det(A⁻¹) = 1\u002Fdet(A)\n  det(cA) = cⁿdet(A)  (n×n matrix)\n  Row swap: det changes sign\n  Row multiply by k: det multiplied by k\n  Add multiple of row to another: det unchanged\n  Triangular matrix: det = product of diagonal entries\n\nGeometric meaning:\n  2×2: |det(A)| = area of parallelogram spanned by rows\u002Fcolumns\n  3×3: |det(A)| = volume of parallelepiped\n  det(A) > 0: orientation preserved\n  det(A) \u003C 0: orientation reversed\n  det(A) = 0: matrix is singular (rows\u002Fcolumns linearly dependent)\n\nInverse formula:\n  A⁻¹ = (1\u002Fdet(A)) · adj(A)\n  adj(A) = (Cᵢⱼ)ᵀ  (adjugate = transpose of cofactor matrix)\n```\n\n---\n\n## Vector Spaces\n```\nVector space V over field F:\n  Satisfies 8 axioms (closure, commutativity, associativity,\n  identity, inverse for + ; associativity, distributivity, identity for ·)\n\nExamples:\n  ℝⁿ: n-tuples of real numbers\n  Mₘₙ: m×n matrices\n  Pₙ: polynomials of degree ≤ n\n  C[a,b]: continuous functions on [a,b]\n  Solution set of Ax = 0 (null space)\n\nSubspace: subset W of V satisfying:\n  0 ∈ W\n  u,v ∈ W → u+v ∈ W\n  u ∈ W → cu ∈ W  (for all scalars c)\n\nFour fundamental subspaces of A (m×n):\n  Column space C(A): span of columns, subspace of ℝᵐ\n  Row space R(A): span of rows, subspace of ℝⁿ\n  Null space N(A): {x: Ax=0}, subspace of ℝⁿ\n  Left null space N(Aᵀ): {y: Aᵀy=0}, subspace of ℝᵐ\n\n  dim(C(A)) = dim(R(A)) = rank(A)  (fundamental theorem)\n  N(A) ⊥ R(A) and N(Aᵀ) ⊥ C(A)\n\nBasis and dimension:\n  Basis: linearly independent spanning set\n  Dimension: number of vectors in any basis\n  Standard basis ℝⁿ: e₁=(1,0,...,0), ..., eₙ=(0,...,0,1)\n  Change of basis: P·[x]_B = [x]_standard\n```\n\n---\n\n## Eigentheory\n```\nEigenvalue equation: Av = λv\n  λ = eigenvalue (scalar)\n  v = eigenvector (nonzero vector)\n  (A - λI)v = 0 → det(A - λI) = 0\n\nCharacteristic polynomial:\n  p(λ) = det(A - λI) = 0\n  Degree n for n×n matrix → n eigenvalues (counting multiplicity)\n  Roots may be complex even for real A\n\nFinding eigenvectors:\n  For each eigenvalue λᵢ: solve (A-λᵢI)v = 0\n  Eigenspace = null space of (A-λᵢI)\n\nProperties:\n  Trace(A) = sum of eigenvalues = Σλᵢ\n  det(A) = product of eigenvalues = Πλᵢ\n  Similar matrices have same eigenvalues\n  Symmetric A: real eigenvalues, orthogonal eigenvectors\n  Symmetric A: positive definite ↔ all eigenvalues > 0\n\nDiagonalization:\n  A = PDP⁻¹  where D = diag(λ₁,...,λₙ)\n  P = matrix of eigenvectors as columns\n  Requires n linearly independent eigenvectors\n  If A has n distinct eigenvalues → diagonalizable\n\nPowers and functions of matrices:\n  Aᵏ = PDᵏP⁻¹  (Dᵏ = diag(λ₁ᵏ,...,λₙᵏ))\n  eᴬ = PeᴰP⁻¹  (eᴰ = diag(eˡ¹,...,eˡⁿ))\n\nSpectral theorem:\n  A symmetric → A = QΛQᵀ  (Q orthogonal, Λ diagonal)\n  Eigenvectors orthonormal, eigenvalues real\n```\n\n---\n\n## Inner Product Spaces\n```\nInner product ⟨u,v⟩:\n  Conjugate symmetry: ⟨u,v⟩ = ⟨v,u⟩* (or symmetric for real)\n  Linearity: ⟨au+bw,v⟩ = a⟨u,v⟩ + b⟨w,v⟩\n  Positive definite: ⟨v,v⟩ ≥ 0, = 0 ↔ v = 0\n\nStandard inner product (dot product): ⟨u,v⟩ = u·v = Σuᵢvᵢ\nFunction inner product: ⟨f,g⟩ = ∫ₐᵇ f(x)g(x)dx\n\nNorm: ||v|| = √⟨v,v⟩\nCauchy-Schwarz: |⟨u,v⟩| ≤ ||u|| ||v||\nAngle: cosθ = ⟨u,v⟩\u002F(||u|| ||v||)\n\nOrthogonality:\n  u ⊥ v ↔ ⟨u,v⟩ = 0\n  Orthogonal set: mutually orthogonal\n  Orthonormal set: orthogonal + each ||vᵢ|| = 1\n  Pythagorean theorem: ||u+v||² = ||u||² + ||v||²  if u⊥v\n\nGram-Schmidt process:\n  Input: {v₁,...,vₙ} linearly independent\n  Output: {u₁,...,uₙ} orthonormal basis\n\n  u₁ = v₁\u002F||v₁||\n  w₂ = v₂ - ⟨v₂,u₁⟩u₁\n  u₂ = w₂\u002F||w₂||\n  wₖ = vₖ - Σᵢ₌₁^(k-1) ⟨vₖ,uᵢ⟩uᵢ\n  uₖ = wₖ\u002F||wₖ||\n\nOrthogonal complement:\n  W⊥ = {v: ⟨v,w⟩=0 ∀w∈W}\n  V = W ⊕ W⊥ (direct sum)\n  Projection: proj_W v = Σᵢ⟨v,uᵢ⟩uᵢ  (orthonormal basis {uᵢ} for W)\n```\n\n---\n\n## Matrix Decompositions\n```python\ndef matrix_decompositions():\n    return {\n        'LU Decomposition': {\n            'form':         'A = LU  (L lower triangular, U upper triangular)',\n            'use':          'Solve Ax=b: Ly=b, then Ux=y',\n            'advantage':    'Reuse factorization for multiple b vectors',\n            'with_pivoting':'PA = LU  (P = permutation, for numerical stability)',\n            'complexity':   'O(n³) once, O(n²) per solve'\n        },\n        'QR Decomposition': {\n            'form':         'A = QR  (Q orthogonal, R upper triangular)',\n            'method':       'Gram-Schmidt, Householder, Givens rotations',\n            'use':          'Least squares, eigenvalue algorithms',\n            'stability':    'More numerically stable than LU for least squares'\n        },\n        'Eigendecomposition': {\n            'form':         'A = PDP⁻¹  (only for diagonalizable A)',\n            'symmetric':    'A = QΛQᵀ  (spectral theorem, Q orthogonal)',\n            'use':          'Matrix powers, differential equations, PCA'\n        },\n        'SVD (Singular Value Decomposition)': {\n            'form':         'A = UΣVᵀ  (any m×n matrix)',\n            'U':            'm×m orthogonal (left singular vectors)',\n            'Σ':            'm×n diagonal (singular values σ₁≥σ₂≥...≥0)',\n            'V':            'n×n orthogonal (right singular vectors)',\n            'rank':         'rank(A) = number of nonzero singular values',\n            'uses': [\n                'Low-rank approximation: Aₖ = UₖΣₖVₖᵀ (best rank-k approximation)',\n                'PCA: SVD of centered data matrix',\n                'Pseudoinverse: A⁺ = VΣ⁺Uᵀ',\n                'Least squares: x = A⁺b',\n                'Image compression, recommender systems, NLP (LSA)'\n            ]\n        },\n        'Cholesky': {\n            'form':         'A = LLᵀ  (A symmetric positive definite)',\n            'advantage':    'Twice as fast as LU, numerically stable',\n            'use':          'SPD systems, multivariate statistics'\n        },\n        'Jordan Normal Form': {\n            'form':         'A = PJP⁻¹  (J = Jordan blocks)',\n            'use':          'Non-diagonalizable matrices, theoretical analysis',\n            'jordan_block':  'Jλ = [[λ 1 0][0 λ 1][0 0 λ]] (λ on diagonal, 1 above)'\n        }\n    }\n```\n\n---\n\n## Least Squares & Applications\n```\nLeast squares problem:\n  Ax = b has no solution (overdetermined, m > n)\n  Find x̂ minimizing ||Ax - b||²\n  Normal equations: AᵀAx̂ = Aᵀb\n  Solution: x̂ = (AᵀA)⁻¹Aᵀb  (if AᵀA invertible)\n  Via SVD: x̂ = A⁺b = VΣ⁺Uᵀb\n\nPCA (Principal Component Analysis):\n  Center data: subtract mean\n  Compute covariance matrix C = (1\u002Fn)XᵀX\n  Eigendecompose: C = QΛQᵀ\n  Or SVD of X: X = UΣVᵀ\n  Principal components = columns of V (or Q)\n  Variance explained by kth PC = σₖ²\u002FΣσᵢ²\n\nLinear transformations:\n  T: ℝⁿ → ℝᵐ is linear if T(u+v)=T(u)+T(v), T(cu)=cT(u)\n  Every linear transformation: T(x) = Ax for some matrix A\n  Kernel (null space): ker(T) = {x: T(x)=0}\n  Image (range): im(T) = {T(x): x∈ℝⁿ} = C(A)\n  Rank-Nullity: dim(ker) + dim(im) = n\n\nQuadratic forms:\n  Q(x) = xᵀAx  (A symmetric)\n  Positive definite: Q(x) > 0 ∀x≠0 ↔ all eigenvalues > 0\n  Negative definite: all eigenvalues \u003C 0\n  Indefinite: mixed eigenvalue signs\n  Classification by eigenvalues of A\n```\n\n---\n\n## Numerical Linear Algebra\n```python\ndef numerical_methods_linalg():\n    return {\n        'Condition number': {\n            'definition':   'κ(A) = ||A|| · ||A⁻¹|| = σ_max\u002Fσ_min',\n            'interpretation': 'How sensitive solution is to perturbations in b',\n            'ill-conditioned': 'κ >> 1: small error in b → large error in x',\n            'well-conditioned': 'κ ≈ 1: stable computation'\n        },\n        'Iterative methods (large sparse A)': {\n            'Jacobi':       'Diagonal update, simple but slow',\n            'Gauss-Seidel': 'Use updated values immediately, faster',\n            'CG (Conjugate Gradient)': 'For SPD matrices, optimal convergence',\n            'GMRES':        'For non-symmetric, Krylov subspace method',\n            'Convergence':  'Depends on eigenvalue distribution and condition number'\n        },\n        'Power iteration (eigenvalues)': {\n            'method':       'Repeatedly multiply by A and normalize',\n            'finds':        'Largest eigenvalue and corresponding eigenvector',\n            'QR algorithm': 'Standard for all eigenvalues, O(n³)',\n            'Lanczos':      'For large sparse symmetric matrices'\n        },\n        'Floating point issues': {\n            'loss_of_significance': 'Subtracting nearly equal numbers',\n            'overflow_underflow':   'Extremely large\u002Fsmall intermediate values',\n            'pivoting':             'Partial pivoting prevents division by small numbers'\n        }\n    }\n```\n\n---\n\n## Key Theorems Summary\n```\nInvertible Matrix Theorem (all equivalent for n×n A):\n  A is invertible ↔\n  det(A) ≠ 0 ↔\n  Ax = b has unique solution for every b ↔\n  Ax = 0 has only trivial solution ↔\n  rank(A) = n ↔\n  columns of A are linearly independent ↔\n  rows of A are linearly independent ↔\n  0 is not an eigenvalue of A ↔\n  AᵀA is invertible ↔\n  A = product of elementary matrices\n\nRank-Nullity: rank(A) + nullity(A) = n  (# columns)\nDimension theorem: dim(C(A)) = dim(R(A)) = rank(A)\nSpectral theorem: Symmetric A → orthogonally diagonalizable\nSVD: Every A = UΣVᵀ (exists for any matrix)\n```\n\n---\n\n## Common Pitfalls\n\n| Pitfall | Fix |\n|---|---|\n| AB = BA assumed | Matrix multiplication is NOT commutative |\n| (A+B)² = A²+2AB+B² | Only if AB = BA; general: (A+B)² = A²+AB+BA+B² |\n| det(A+B) = det(A)+det(B) | WRONG: determinant is NOT linear in this way |\n| Eigenvalues of AB = product of eigenvalues | Only true if A and B share eigenvectors |\n| Rank = number of nonzero rows | Rank = number of nonzero rows in REF |\n| Orthogonal = perpendicular only | Orthogonal matrix has columns forming orthonormal set |\n\n---\n\n## Related Skills\n\n- **calculus-expert**: Multivariable and vector calculus\n- **statistics-expert**: PCA, regression, covariance matrices\n- **quantum-mechanics-expert**: Linear operators in Hilbert space\n- **machine-learning-expert**: SVD, PCA, neural networks\n- **numerical-methods-expert**: Numerical linear algebra\n- **differential-equations-expert**: Systems of ODEs using matrices\n",{"name":5,"author":44,"license":15,"metadata":45,"description":7},"luo-kai",{"author":46,"version":47,"category":28},"luokai25","1.0","main","4d2fb89cdfb75f3f6fd3c6c5ec9ad90ea129eb3c","ai-agent-skills\u002F13-physics-and-mathematics (by Luo 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Kai)\u002Flinear-algebra-expert\u002F.gitkeep",{"path":77,"resourceType":72,"resourceTypeDesc":73,"sizeBytes":78,"sourceHtmlUrl":79},"customize.toml",2371,"https:\u002F\u002Fgithub.com\u002FdevelopAgent\u002Fbmad-start\u002Fblob\u002Fmain\u002F.agent\u002Fskills\u002Fgds-agent-game-solo-dev\u002Fcustomize.toml",[81,108,125,155,187,224],{"skillNo":82,"name":83,"nameZh":6,"description":84,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":85,"categories":86,"keywords":87,"publishedAt":39},"01M6F2SKTE208174C7S7JJCZC7","write-contracts","Generates secure Aptos Move V2 smart contracts with Object model, Digital Asset integration, security patterns, and storage type guidance. Includes comprehensive storage decision framework for optimal data structure selection. Triggers on: 'write contract', 'create NFT collection', 'build marketplace', 'implement minting', 'generate Move module', 'create token contract', 'build DAO', 'implement staking'. Ask storage questions when: 'store', 'track', 'registry', 'mapping', 'list', 'collection'.","https:\u002F\u002Fgithub.com\u002Fluokai0\u002Fai-agent-skills-by-luo-kai\u002Fblob\u002Fmain\u002Fai-agent-skills\u002F22-clawhub-skills\u002Fwrite-contracts\u002FSKILL.md",[],[88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107],"write","contracts","write contracts","storage","secure","create","generate","collection","model","decision","ask","smart-contracts","selection","objects","structure","nft","token","aptos","critical","move",{"skillNo":109,"name":110,"nameZh":6,"description":111,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":112,"categories":113,"keywords":114,"publishedAt":39},"01M6F2RFXJ6T9MKW0A7N7GM45B","web-search","通用网络搜索技能，支持多引擎搜索（百度、必应、DuckDuckGo），无需API密钥即可获取实时信息","https:\u002F\u002Fgithub.com\u002Fluokai0\u002Fai-agent-skills-by-luo-kai\u002Fblob\u002Fmain\u002Fai-agent-skills\u002F22-clawhub-skills\u002Fweb-search-ex-skill\u002FSKILL.md",[],[115,116,117,118,119,120,121,122,123,124],"web","search","web search","搜索","baidu","bing","duckduckgo","百度","多引擎","internet",{"skillNo":126,"name":127,"nameZh":6,"description":128,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":129,"categories":130,"keywords":136,"publishedAt":39},"01M6F2K968G1FZEZFPW2E6B222","solo-build","Execute implementation plan tasks with TDD workflow, auto-commit, and phase gates. 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Covers welcome emails, password resets, notifications, order confirmations, newsletters, transactional emails, and the embeddable email editor component.","https:\u002F\u002Fgithub.com\u002Fluokai0\u002Fai-agent-skills-by-luo-kai\u002Fblob\u002Fmain\u002Fai-agent-skills\u002F22-clawhub-skills\u002Freact-email-skills\u002FSKILL.md",[161,164],{"categoryNo":162,"categoryName":163,"taxonomyType":134,"taxonomyTypeDesc":135},"01JBSEEDCAT000000000000005","邮件与沟通",{"categoryNo":165,"categoryName":166,"taxonomyType":134,"taxonomyTypeDesc":135},"01JBSEEDCAT000000000000009","前端开发",[168,169,170,171,172,173,174,175,176,177,178,179,180,27,181,182,183,184,185,186],"react","email","react email","github","editor","components","react components","component","welcome","https","notifications","password","node","html","resend","visual","emails","rendering","with",{"skillNo":188,"name":189,"nameZh":6,"description":190,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":191,"categories":192,"keywords":205,"publishedAt":39},"01M6F2FBVGVDFSG8SSFJGXA40C","APITester Agent-Driven API Testing","Test API endpoints and document responses. Define tests in plain English, run them, get formatted results. 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Creates prioritized, sequenced cleanup processes with effort estimates, dependencies, and automation feasibility. Use after running \u002Fhubspot-audit.","https:\u002F\u002Fgithub.com\u002Fluokai0\u002Fai-agent-skills-by-luo-kai\u002Fblob\u002Fmain\u002Fai-agent-skills\u002F22-clawhub-skills\u002Fhubspot-implementation-plan\u002FSKILL.md",[],[146,231,232,233,234,235,236,94,237,238,239,240,241,27,242,243,186,244],"implementation","hubspot","hubspot implementation plan","automation","feasibility","cleanup","planning","report","audit","audit report","phased implementation","dependencies","processes","hubspot audit"]