linear-algebra-ai999x — offline, verified linear algebra solver with step-by-step answers

Install · Capability guide · Evaluation · AI999X on Hugging Face · Website · Support: Buy Me a Coffee

License: Apache-2.0 · Author: Ibnul Jaif Farabi (creator and engineer) · Organization: AI999X · Version 1.0.3

Field: Mathematics & Statistics · Tasks: Question Answering, Token Classification, Text Classification, Sentence Similarity, Feature Extraction

linear-algebra-ai999x is an offline, deterministic Python solver that answers linear algebra questions with verified results from 89 certified capabilities, gives 399 documented answers identically on Windows and Linux, and refuses rather than guesses when an answer cannot be verified.

linear-algebra-ai999x is a compact, deterministic linear algebra engine for Python. It reads a question in plain English (or a typed call), picks a certified linear algebra capability, computes the answer exactly, checks it, and returns the result with its steps, units, LaTeX and assumptions — or an honest refusal that says what is missing. It runs fully offline on any CPU, needs no GPU, no API key and no internet, and is built to give zero wrong answers: when it is not sure, it refuses instead of guessing.

Highlights

• Verified, not generated. Every answer is computed by a certified linear algebra capability and checked (substituted back, dimension-checked or compared with an independent computation) before it is returned.
• Zero wrong answers by design. Seven explicit outcomes — SOLVED, NEEDS_INPUT, AMBIGUOUS, INVALID_INPUT, OUT_OF_SCOPE, UNRECOGNIZED_FORMAT, INTERNAL_ERROR — never a silent guess. Every one of its 399 documented answers is reproduced exactly on every tested platform.
• Plain-English questions and a guaranteed typed interface. ask() reads documented question forms; solve(capability_id, **inputs) always solves a supported task with valid inputs.
• Explains itself. Steps, formulas in LaTeX, units, assumptions and the model used, ready for notebooks, tutoring apps and homework checkers.
• Small and portable. Pure Python (sympy, mpmath); identical answers on Windows and Linux (Python 3.10, 3.11, 3.12, 3.13, 3.14).
• Coverage: vectors, matrices, determinants, linear systems, eigenvalues, decompositions, least squares, sparse matrices and numerical methods (root finding, interpolation, quadrature).

Model overview

Property	Value
Package	linear-algebra-ai999x 1.0.3 (import linear_algebra_ai999x)
Type	Deterministic neuro-symbolic linear algebra engine (rule-based reading + certified symbolic capabilities)
Capabilities	89 certified procedures
Knowledge pack	version 1.0.0
Golden answers	399 documented answers, identical on every tested platform
Platforms tested	Windows (Python 3.13), Linux x86-64 (Python 3.10, 3.11, 3.12, 3.13, 3.14)
Runtime	CPU only, offline; no GPU, no network, no API key
License	Apache-2.0
Creator / engineer	Ibnul Jaif Farabi
Organization	AI999X

Quickstart

pip install linear-algebra-ai999x
# or straight from this repository:
pip install https://huggingface.co/AI999X/linear-algebra-ai999x/resolve/main/dist/linear_algebra_ai999x-1.0.3-py3-none-any.whl

import linear_algebra_ai999x as linear_algebra

linear_algebra.ask("Find the determinant of [[2, 0, 1], [1, 3, 2], [1, 1, 2]].").show()
# -> SOLVED — det = 6  (determinant_3x3)
linear_algebra.ask("Solve the system 2x + 3y = 7, x - y = 1.").show()
# -> SOLVED — solution = [2.0, 1.0], det = -5  (solve_2x2_system)
linear_algebra.ask("Find the eigenvalues of [[2, 1], [1, 2]].").show()
# -> SOLVED — lambda1 = 1, lambda2 = 3  (eigenvalues_2x2)
linear_algebra.ask("Use Newton's method to find a root of x^3 - 2x - 5 starting from x0 = 2.").show()
# -> SOLVED — root = 2.09455  (newton_root)
linear_algebra.solve("qr_decomposition", A="[[3, 1], [4, 2]]").show()
# -> SOLVED — Q = [[0.5999999999999999, -0.7999999999999999], [0.7999999999999999, 0.6000000000000001]], R = [[5.0, 2.1999999999999997], [-0.0, 0.40000000000000013]]  (qr_decomposition)
linear_algebra.ask("Who created you?").message
# -> linear-algebra-ai999x was created and engineered by Ibnul Jaif Farabi at AI999X.

Command line: linear-algebra-ai999x "<question>", linear-algebra-ai999x --capabilities. The outputs above were produced by running this exact code on linear-algebra-ai999x 1.0.3.

Find the right capability, and draw it

linear_algebra.search() finds which certified capability a question is about — across every AI999X model, so a question for another model points you to it. It runs ai999x-encoder, AI999X's own capability encoder (4.7M parameters, a 4.9 MB ONNX model, offline; the root model.safetensors of this repository): pip install "linear-algebra-ai999x[search]". ask(question, suggest=True) adds the same pointers to a question it does not answer; an answer itself never changes.

import linear_algebra_ai999x as linear_algebra

linear_algebra.search("Solve x^2 - 5x + 6 = 0", k=2)
# -> Roots of a quadratic equation (math-foundations-ai999x, 0.63); Roots of a cubic equation (math-foundations-ai999x, 0.59)

How every answer is verified

1. Reading — the question is turned into typed quantities (value, unit, kind, role) and one asked quantity; every stated number must be used or the question is refused.
2. Planning — only certified capabilities may answer; a capability applies only under the conditions the wording states (for example a pendulum, a liquid at rest, boiling).
3. Computing — exact symbolic or high-precision numeric evaluation.
4. Checking — dimensions, substitution back into the defining equation, and agreement between independent readings; any disagreement becomes a refusal, never a number.

Capabilities are certified before they ship: each must reproduce hand-worked reference cases, reject a deliberately wrong variant (a control), and leave every benchmark answer unchanged.

Worked examples

Each question was put to linear-algebra-ai999x; the answer is the one it returned, with the steps it took.

Determinant of a 3 x 3 matrix · Determinants

Find the determinant of [[2, 0, 1], [1, 3, 2], [1, 1, 2]].

→ det = 6

Steps

• Determinant of a 3 x 3 matrix: A = [2 0 1; 1 3 2; 1 1 2] → det = 6

Two linear equations in two unknowns · Linear systems

Solve the system 2x + 3y = 7, x - y = 1.

→ solution = [2, 1], det = -5

Steps

• Two linear equations in two unknowns: A = [2 3; 1 -1], b = [7, 1] → solution = [2, 1], det = -5

Eigenvalues of a 2 x 2 matrix · Eigenvalues

Find the eigenvalues of [[2, 1], [1, 2]].

→ lambda1 = 1, lambda2 = 3

Steps

• Eigenvalues of a 2 x 2 matrix: A = [2 1; 1 2] → lambda1 = 1, lambda2 = 3

Root of a polynomial by Newton's method · Numerical methods

Use Newton's method to find a root of x^3 - 2x - 5 starting from x0 = 2.

→ root = 2.094551482

Steps

• Root of a polynomial by Newton's method: coefficients = [1, 0, -2, -5], x0 = 2 → root = 2.094551482

QR decomposition (Householder) · Decompositions

m.solve("qr_decomposition", A="[[3, 1], [4, 2]]")

→ Q = [[0.6, -0.8], [0.8, 0.6]], R = [[5, 2.2], [-0, 0.4]]

Steps

• QR decomposition (Householder): A = [3 1; 4 2] → Q = [[0.6, -0.8], [0.8, 0.6]], R = [[5, 2.2], [-0, 0.4]]

Least-squares regression line · Least squares

Find the least squares line for the points (1, 2), (2, 3), (3, 5), (4, 4).

→ slope = 0.8, intercept = 1.5, r_squared = 0.64

Steps

• Least-squares regression line: x = [1, 2, 3, 4], y = [2, 3, 5, 4] → slope = 0.8, intercept = 1.5, r_squared = 0.64

Figures

Every point was computed by linear-algebra-ai999x.

A linear map of the plane and its eigenvectors

The unit square and its image under A = [[2, 1], [1, 2]] (each corner mapped by the package's matrix–vector product), with the eigenvectors it returned and their images: λ1 = 1 along (0.707107, -0.707107); λ2 = 3 along (0.707107, 0.707107).

The unit circle under a matrix: singular values as semi-axes

72 points of the unit circle mapped by A = [[3, 0], [4, 5]]: an ellipse whose semi-axes are the singular values σ₁ = 6.7082 and σ₂ = 2.23607 returned by the package (the longest and shortest mapped radii drawn: 6.7082 and 2.23607).

LU factorisation as matrices

The LU factorisation with partial pivoting returned by the package for A = [[2, 1, 1], [4, -6, 0], [-2, 7, 2]]: A = P L U with P a permutation, L unit lower triangular and U upper triangular (cells shaded by magnitude; negative values in the second colour). The product P L U, also computed by the package, equals A.

Least-squares polynomial and line through the same points

Least-squares fits to the points x = 0, 1, 2, 3, y = 1, 0, 3, 10: the degree-2 polynomial y = 1 − 3x + 2x² (coefficients from the package, the curve evaluated by it at 61 points) and the straight line y = −1 + 3x with R² = 0.737705.

Interpolation through five nodes: spline, polynomial, linear

Three interpolants through the nodes x = 0, 1, 2, 3, 4, y = 0, 2, 1, 3, 2, each evaluated by the package at 81 points: the natural cubic spline, the degree-4 Lagrange polynomial and piecewise-linear interpolation; marked: x = 2.5, spline value = 1.91964.

Capabilities

89 certified procedures. A few of them:

• Procedures: 1-D Poisson equation by finite differences; 2-D Poisson equation on a grid (5-point stencil); Adjugate (classical adjoint) of a matrix; Angle between two vectors; Area of the parallelogram (and triangle) spanned by two vectors; Best rank-k approximation (Eckart-Young); Characteristic polynomial of a matrix; Cholesky decomposition; Clamped cubic spline interpolation; Coefficients of the interpolating polynomial; Column space of a matrix; Composite midpoint rule for a polynomial; Compressed sparse column (CSC) form of a matrix; Compressed sparse row (CSR) form of a matrix; Condition number of a matrix; Conjugate gradient method for Ax = b; Coordinates of a vector in a basis; Cross product of two 3-D vectors; Dense matrix from CSR arrays; Derivative of equally spaced data (finite differences); Determinant of a 2 x 2 matrix; Determinant of a 3 x 3 matrix; Determinant of an n x n matrix; Difference of two matrices …

linear_algebra_ai999x.templates() lists every question form ask() reads; the capability guide (PDF, every example executed) is included in this repository.

All 89 formulas

Show every formula, by topic

Decompositions / Cholesky

Cholesky decomposition

$$A=L{L}^{T}A\; =\; LL^\{T\}$$

Decompositions / LU

LU decomposition with partial pivoting

$$A=PLUA\; =\; PLU$$

Decompositions / QR

QR decomposition (Householder)

$$A=QR,Q^{T}Q=IA\; =\; QR,\backslash \; Q^\{T\}Q\; =\; I$$

Decompositions / polar decomposition

Polar decomposition

$$A=UP,P=(A^{T}A{)}^{1\mathrm{/}2},U=A{P}^{-1}A\; =\; UP,\backslash quad\; P\; =\; (A^\{T\}A)^\{1/2\},\backslash \; U\; =\; AP^\{-1\}$$

Decompositions / singular value decomposition

Best rank-k approximation (Eckart-Young)

$$A_k=\sum _{i=1}^k{\sigma }_i{\mathbf{u}}_i{\mathbf{v}}_i^{T}A\_k\; =\; \backslash sum\_\{i=1\}^\{k\}\; \backslash sigma\_i\; \backslash mathbf\{u\}\_i\; \backslash mathbf\{v\}\_i^\{T\}$$

Singular values of a matrix

$$A=U\mathrm{\Sigma }{V}^T,{\sigma }_i=\sqrt{{\lambda }_i(A^{T}A)}A\; =\; U\backslash Sigma\; V^\{T\},\backslash quad\; \backslash sigma\_i\; =\; \backslash sqrt\{\backslash lambda\_i(A^\{T\}A)\}$$

Determinants / cofactor expansion

Adjugate (classical adjoint) of a matrix

$$\mathrm{adj}(A{)}_{ij}=C_{ji}\backslash operatorname\{adj\}(A)\_\{ij\}\; =\; C\_\{ji\}$$

Minor and cofactor of an entry

$$C_{ij}=(-1{)}^{i+j}{M}_{ij}C\_\{ij\}\; =\; (-1)^\{i+j\}\; M\_\{ij\}$$

Determinants / general determinants

Determinant of an n x n matrix

$$\det A=(-1{)}^s\prod _i{u}_{ii}\backslash det\; A\; =\; (-1)^\{s\}\backslash prod\_i\; u\_\{ii\}$$

Determinants / small determinants

Determinant of a 2 x 2 matrix

Determinant of a 3 x 3 matrix

$$\det A=a_{11}{a}_{22}{a}_{33}+a_{12}{a}_{23}{a}_{31}+a_{13}{a}_{21}{a}_{32}-a_{13}{a}_{22}{a}_{31}-a_{11}{a}_{23}{a}_{32}-a_{12}{a}_{21}{a}_{33}\backslash det\; A\; =\; a\_\{11\}a\_\{22\}a\_\{33\}\; +\; a\_\{12\}a\_\{23\}a\_\{31\}\; +\; a\_\{13\}a\_\{21\}a\_\{32\}\; -\; a\_\{13\}a\_\{22\}a\_\{31\}\; -\; a\_\{11\}a\_\{23\}a\_\{32\}\; -\; a\_\{12\}a\_\{21\}a\_\{33\}$$

Eigenvalues / characteristic polynomial

Characteristic polynomial of a matrix

$$p(\lambda )=\det (\lambda I-A)p(\backslash lambda)\; =\; \backslash det(\backslash lambda\; I\; -\; A)$$

Eigenvalues / definiteness

Positive definiteness test (Sylvester's criterion)

$$A\succ 0⟺D_k=\det A_{[\mathrm{1..}k],[\mathrm{1..}k]}>0,k=\mathrm{1..}nA\; \backslash succ\; 0\; \backslash iff\; D\_k\; =\; \backslash det\; A\_\{[1..k],[1..k]\}\; >\; 0,\backslash \; k\; =\; 1..n$$

Eigenvalues / general matrices

Eigenvalues of a general square matrix (QR algorithm)

$$\det (A-\lambda I)=0,\rho (A)=\underset{i}{\max }\mathrm{\mid }{\lambda }_i\mathrm{\mid }\backslash det(A\; -\; \backslash lambda\; I)\; =\; 0,\backslash quad\; \backslash rho(A)\; =\; \backslash max\_i\; \backslash vert\; \backslash lambda\_i\backslash vert$$

Eigenvalues / iterative methods

Dominant eigenvalue by power iteration

$${\mathbf{x}}_{k+1}=\frac{A{\mathbf{x}}_k}{vertA{\mathbf{x}}_{k}vert},\lambda \approx {\mathbf{x}}_k^{T}A{\mathbf{x}}_k\backslash mathbf\{x\}\_\{k+1\}\; =\; \backslash frac\{A\backslash mathbf\{x\}\_k\}\{\backslash \backslash vert\; A\backslash mathbf\{x\}\_k\backslash \backslash vert\; \},\backslash quad\; \backslash lambda\; \backslash approx\; \backslash mathbf\{x\}\_k^\{T\}A\backslash mathbf\{x\}\_k$$

Smallest eigenvalue by inverse iteration

$$A{\mathbf{y}}_{k+1}={\mathbf{x}}_k,{\mathbf{x}}_{k+1}={\mathbf{y}}_{k+1}\mathrm{/}vert{\mathbf{y}}_{k+1}vertA\backslash mathbf\{y\}\_\{k+1\}\; =\; \backslash mathbf\{x\}\_k,\backslash quad\; \backslash mathbf\{x\}\_\{k+1\}\; =\; \backslash mathbf\{y\}\_\{k+1\}/\backslash \backslash vert\; \backslash mathbf\{y\}\_\{k+1\}\backslash \backslash vert$$

Eigenvalues / small matrices

Eigenvalues of a 2 x 2 matrix

$$\lambda =\frac{\mathrm{tr}A\pm \sqrt{(\mathrm{tr}A{)}^2-4\det A}}{2}\backslash lambda\; =\; \backslash frac\{\backslash operatorname\{tr\}A\; \backslash pm\; \backslash sqrt\{(\backslash operatorname\{tr\}A)^2\; -\; 4\backslash det\; A\}\}\{2\}$$

Eigenvalues / symmetric matrices

Eigenvalues and eigenvectors of a symmetric matrix

$$A{\mathbf{q}}_i={\lambda }_i{\mathbf{q}}_i,Q^{T}Q=IA\backslash mathbf\{q\}\_i\; =\; \backslash lambda\_i\; \backslash mathbf\{q\}\_i,\backslash \; Q^\{T\}Q\; =\; I$$

Eigenvalues of a symmetric matrix (Jacobi method)

$$A=Q\mathrm{\Lambda }{Q}^{T}A\; =\; Q\backslash Lambda\; Q^\{T\}$$

Least squares / line fitting

Least-squares regression line

$$b=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x}{)}^2},a=\bar{y}-b\bar{x}b\; =\; \backslash frac\{\backslash sum\; (x\_i-\backslash bar\; x)(y\_i-\backslash bar\; y)\}\{\backslash sum\; (x\_i-\backslash bar\; x)^2\},\backslash quad\; a\; =\; \backslash bar\; y\; -\; b\backslash bar\; x$$

Least squares / orthogonal projections

Orthogonal projection matrix onto a column space

$$P=A(A^{T}A{)}^{-1}{A}^{T}P\; =\; A(A^\{T\}A)^\{-1\}A^\{T\}$$

Least squares / overdetermined systems

Least-squares solution of an overdetermined system

$$A^{T}A\widehat{\mathbf{x}}=A^T\mathbf{b}A^\{T\}A\backslash ,\backslash hat\{\backslash mathbf\{x\}\}\; =\; A^\{T\}\backslash mathbf\{b\}$$

Least squares / polynomial fitting

Least-squares polynomial fit

$$\underset{\mathbf{c}}{\min }\sum _i(y_i-\sum _{k=0}^d{c}_k{x}_i^k{)}^2\backslash min\_\{\backslash mathbf\{c\}\}\; \backslash sum\_i\; \backslash Big(y\_i\; -\; \backslash sum\_\{k=0\}^\{d\}\; c\_k\; x\_i^k\backslash Big)^2$$

Least squares / pseudoinverse

Moore-Penrose pseudoinverse (full rank)

$$A^{+}=(A^{T}A{)}^{-1}{A}^T\text{ or }{A}^T(A{A}^T{)}^{-1}A^\{+\}\; =\; (A^\{T\}A)^\{-1\}A^\{T\}\backslash \; \backslash text\{or\}\backslash \; A^\{T\}(AA^\{T\})^\{-1\}$$

Linear systems / Gaussian elimination

Solve a square linear system Ax = b

$$A\mathbf{x}=\mathbf{b}A\backslash mathbf\{x\}\; =\; \backslash mathbf\{b\}$$

Linear systems / banded and sparse systems

Tridiagonal system (Thomas algorithm)

$$a_i{x}_{i-1}+b_i{x}_i+c_i{x}_{i+1}=d_{i}a\_i\; x\_\{i-1\}\; +\; b\_i\; x\_i\; +\; c\_i\; x\_\{i+1\}\; =\; d\_i$$

Linear systems / existence and uniqueness

Number of solutions of a linear system

Linear systems / iterative methods

Conjugate gradient method for Ax = b

$${\alpha }_k=\frac{{\mathbf{r}}_k^T{\mathbf{r}}_k}{{\mathbf{p}}_k^{T}A{\mathbf{p}}_k},{\mathbf{x}}_{k+1}={\mathbf{x}}_k+{\alpha }_k{\mathbf{p}}_k,{\mathbf{p}}_{k+1}={\mathbf{r}}_{k+1}+\frac{{\mathbf{r}}_{k+1}^T{\mathbf{r}}_{k+1}}{{\mathbf{r}}_k^T{\mathbf{r}}_k}{\mathbf{p}}_k\backslash alpha\_k\; =\; \backslash frac\{\backslash mathbf\{r\}\_k^\{T\}\backslash mathbf\{r\}\_k\}\{\backslash mathbf\{p\}\_k^\{T\}A\backslash mathbf\{p\}\_k\},\backslash \; \backslash mathbf\{x\}\_\{k+1\}\; =\; \backslash mathbf\{x\}\_k\; +\; \backslash alpha\_k\backslash mathbf\{p\}\_k,\backslash \; \backslash mathbf\{p\}\_\{k+1\}\; =\; \backslash mathbf\{r\}\_\{k+1\}\; +\; \backslash frac\{\backslash mathbf\{r\}\_\{k+1\}^\{T\}\backslash mathbf\{r\}\_\{k+1\}\}\{\backslash mathbf\{r\}\_k^\{T\}\backslash mathbf\{r\}\_k\}\backslash mathbf\{p\}\_k$$

Gauss-Seidel iteration for Ax = b

Jacobi iteration for Ax = b

$$x_i^{(k+1)}=\frac{1}{a_{ii}}(b_i-\sum _{j\mathrm{\ne }i}{a}_{ij}{x}_j^{(k)})x\_i^\{(k+1)\}\; =\; \backslash frac\{1\}\{a\_\{ii\}\}\backslash Big(b\_i\; -\; \backslash sum\_\{j\backslash ne\; i\}\; a\_\{ij\}\; x\_j^\{(k)\}\backslash Big)$$

Linear systems / row reduction

Reduced row echelon form

$$A\sim \mathrm{rref}(A)A\; \backslash sim\; \backslash operatorname\{rref\}(A)$$

Linear systems / small systems

Three linear equations in three unknowns

$$x_j=\frac{\det A_j}{\det A}x\_j\; =\; \backslash frac\{\backslash det\; A\_j\}\{\backslash det\; A\}$$

Two linear equations in two unknowns

Linear transformations / matrix-vector product

Matrix times vector (image under a linear map)

$$(A\mathbf{x}{)}_i=\sum _j{a}_{ij}{x}_j(A\backslash mathbf\{x\})\_i\; =\; \backslash sum\_j\; a\_\{ij\}\; x\_j$$

Linear transformations / transformations of the plane

Rotation of a plane vector

Matrices / inverses

Inverse of a square matrix

$$A{A}^{-1}=A^{-1}A=IA\; A^\{-1\}\; =\; A^\{-1\}\; A\; =\; I$$

Matrices / matrix arithmetic

Difference of two matrices

$$(A-B{)}_{ij}=a_{ij}-b_{ij}(A-B)\_\{ij\}\; =\; a\_\{ij\}\; -\; b\_\{ij\}$$

Kronecker (tensor) product of two matrices

$$A\otimes B=[a_{ij}B]A\backslash otimes\; B\; =\; [a\_\{ij\}B]$$

Power of a square matrix

$$A^k=\underset{k}{\underbrace{AA\cdots A}}A^k\; =\; \backslash underbrace\{A\; A\; \backslash cdots\; A\}\_\{k\}$$

Product of two matrices

$$(AB{)}_{ij}=\sum _k{a}_{ik}{b}_{kj}(AB)\_\{ij\}\; =\; \backslash sum\_k\; a\_\{ik\}\; b\_\{kj\}$$

Scalar multiple of a matrix

$$(kA{)}_{ij}=k{a}_{ij}(kA)\_\{ij\}\; =\; k\backslash ,a\_\{ij\}$$

Sum of two matrices

$$(A+B{)}_{ij}=a_{ij}+b_{ij}(A+B)\_\{ij\}\; =\; a\_\{ij\}\; +\; b\_\{ij\}$$

Trace of a square matrix

$$\mathrm{tr}A=\sum _i{a}_{ii}\backslash operatorname\{tr\}A\; =\; \backslash sum\_i\; a\_\{ii\}$$

Transpose of a matrix

$$(A^T{)}_{ij}=a_{ji}(A^\{T\})\_\{ij\}\; =\; a\_\{ji\}$$

Matrices / norms and conditioning

Condition number of a matrix

$${\kappa }_2(A)=\frac{{\sigma }_{\max }}{{\sigma }_{\min }},{\kappa }_1(A)=vertAver{t}_{1}vert{A}^{-1}ver{t}_1\backslash kappa\_2(A)\; =\; \backslash frac\{\backslash sigma\_\{\backslash max\}\}\{\backslash sigma\_\{\backslash min\}\},\backslash quad\; \backslash kappa\_1(A)\; =\; \backslash \backslash vert\; A\backslash \backslash vert\; \_1\backslash \backslash vert\; A^\{-1\}\backslash \backslash vert\; \_1$$

Norms of a matrix

$$vertAver{t}_F=\sqrt{\sum a_{ij}^2},vertAver{t}_1=\underset{j}{\max }{\sum }_i\mathrm{\mid }{a}_{ij}\mathrm{\mid },vertAver{t}_{\mathrm{\infty }}={\max }_i{\sum }_j\mathrm{\mid }{a}_{ij}\mathrm{\mid },vertAver{t}_2={\sigma }_{\max }\backslash \backslash vert\; A\backslash \backslash vert\; \_F\; =\; \backslash sqrt\{\backslash textstyle\backslash sum\; a\_\{ij\}^2\},\backslash \; \backslash \backslash vert\; A\backslash \backslash vert\; \_1\; =\; \backslash max\_j\; \backslash textstyle\backslash sum\_i\; \backslash vert\; a\_\{ij\}\backslash vert\; ,\backslash \; \backslash \backslash vert\; A\backslash \backslash vert\; \_\backslash infty\; =\; \backslash max\_i\; \backslash textstyle\backslash sum\_j\; \backslash vert\; a\_\{ij\}\backslash vert\; ,\backslash \; \backslash \backslash vert\; A\backslash \backslash vert\; \_2\; =\; \backslash sigma\_\{\backslash max\}$$

Matrices / rank

Rank of a matrix

$$\mathrm{rank}A=\dim \mathrm{col}A\backslash operatorname\{rank\}A\; =\; \backslash dim\backslash operatorname\{col\}A$$

Matrix functions / exponential

Matrix exponential

$$e^A=\sum _{k=0}^{\mathrm{\infty }}\frac{A^k}{k!}=(e^{A\mathrm{/}{2}^s}{)}^{2^s}e^\{A\}\; =\; \backslash sum\_\{k=0\}^\{\backslash infty\}\; \backslash frac\{A^k\}\{k!\}\; =\; \backslash big(e^\{A/2^s\}\backslash big)^\{2^s\}$$

Matrix functions / square root

Square root of a positive definite matrix

$$A^{1\mathrm{/}2}=Q{\mathrm{\Lambda }}^{1\mathrm{/}2}{Q}^{T}A^\{1/2\}\; =\; Q\backslash ,\backslash Lambda^\{1/2\}\; Q^\{T\}$$

Numerical PDE / discrete Poisson problems

1-D Poisson equation by finite differences

$$\frac{-u_{k-1}+2u_k-u_{k+1}}{h^2}=f_k,u_0=u_{n+1}=0\backslash frac\{-u\_\{k-1\}\; +\; 2u\_k\; -\; u\_\{k+1\}\}\{h^2\}\; =\; f\_k,\backslash quad\; u\_0\; =\; u\_\{n+1\}\; =\; 0$$

2-D Poisson equation on a grid (5-point stencil)

$$\frac{4u_{i,j}-u_{i-1,j}-u_{i+1,j}-u_{i,j-1}-u_{i,j+1}}{h^2}=f_{i,j}\backslash frac\{4u\_\{i,j\}\; -\; u\_\{i-1,j\}\; -\; u\_\{i+1,j\}\; -\; u\_\{i,j-1\}\; -\; u\_\{i,j+1\}\}\{h^2\}\; =\; f\_\{i,j\}$$

Numerical methods / differentiation

Derivative of equally spaced data (finite differences)

$$y_i^{\mathrm{\prime }}\approx \frac{y_{i+1}-y_{i-1}}{2h}y\text{'}\_i\; \backslash approx\; \backslash frac\{y\_\{i+1\}\; -\; y\_\{i-1\}\}\{2h\}$$

Numerical methods / interpolation

Coefficients of the interpolating polynomial

$$\sum _k{c}_k{x}_i^k=y_i\backslash sum\_k\; c\_k\; x\_i^k\; =\; y\_i$$

Lagrange interpolation

$$p(t)=\sum _i{y}_i\prod _{j\mathrm{\ne }i}\frac{t-x_j}{x_i-x_j}p(t)\; =\; \backslash sum\_i\; y\_i\; \backslash prod\_\{j\backslash ne\; i\}\backslash frac\{t\; -\; x\_j\}\{x\_i\; -\; x\_j\}$$

Newton divided differences

$$f[x_i,\dots ,x_{i+k}]=\frac{f[x_{i+1},\dots ,x_{i+k}]-f[x_i,\dots ,x_{i+k-1}]}{x_{i+k}-x_i}f[x\_i,\backslash ldots,x\_\{i+k\}]\; =\; \backslash frac\{f[x\_\{i+1\},\backslash ldots,x\_\{i+k\}]\; -\; f[x\_i,\backslash ldots,x\_\{i+k-1\}]\}\{x\_\{i+k\}\; -\; x\_i\}$$

Piecewise linear interpolation

$$y=y_k+(y_{k+1}-y_k)\frac{t-x_k}{x_{k+1}-x_k}y\; =\; y\_k\; +\; (y\_\{k+1\}-y\_k)\backslash frac\{t\; -\; x\_k\}\{x\_\{k+1\}\; -\; x\_k\}$$

Numerical methods / polynomials

Evaluate a polynomial and its derivative (Horner's method)

$$p(x)=(\cdots ((a_{n}x+a_{n-1})x+a_{n-2})x+\cdots )x+a_{0}p(x)\; =\; (\backslash cdots((a\_n\; x\; +\; a\_\{n-1\})x\; +\; a\_\{n-2\})x\; +\; \backslash cdots)x\; +\; a\_0$$

Numerical methods / quadrature

Composite midpoint rule for a polynomial

$${\int }_a^{b}p\approx h\sum _{k=0}^{n-1}p(a+(k+\frac{1}{2})h)\backslash int\_a^b\; p\; \backslash approx\; h\backslash sum\_\{k=0\}^\{n-1\}\; p\backslash big(a\; +\; (k+\backslash tfrac12)h\backslash big)$$

Gauss-Legendre quadrature nodes and weights

$${\int }_{-1}^{1}f(x)dx\approx \sum _{i=1}^n{w}_{i}f(x_i),P_n(x_i)=0\backslash int\_\{-1\}^\{1\}\; f(x)\backslash ,dx\; \backslash approx\; \backslash sum\_\{i=1\}^\{n\}\; w\_i\; f(x\_i),\backslash quad\; P\_n(x\_i)\; =\; 0$$

Gauss-Legendre quadrature of a polynomial

$${\int }_a^{b}p(x)dx\approx \frac{b-a}{2}\sum _i{w}_{i}p(\frac{a+b}{2}+\frac{b-a}{2}{x}_i)\backslash int\_a^b\; p(x)\backslash ,dx\; \backslash approx\; \backslash frac\{b-a\}\{2\}\backslash sum\_i\; w\_i\backslash ,p\backslash Big(\backslash frac\{a+b\}\{2\}\; +\; \backslash frac\{b-a\}\{2\}x\_i\backslash Big)$$

Simpson's rule for tabulated data

$$\int ydx\approx \frac{h}{3}(y_0+4\sum _{\text{odd}}{y}_i+2\sum _{\text{even}}{y}_i+y_n)\backslash int\; y\backslash ,dx\; \backslash approx\; \backslash frac\{h\}\{3\}\backslash Big(y\_0\; +\; 4\backslash sum\_\{\backslash text\{odd\}\}\; y\_i\; +\; 2\backslash sum\_\{\backslash text\{even\}\}\; y\_i\; +\; y\_n\backslash Big)$$

Trapezoidal rule for tabulated data

$$\int ydx\approx \sum _i\frac{x_{i+1}-x_i}{2}(y_i+y_{i+1})\backslash int\; y\backslash ,dx\; \backslash approx\; \backslash sum\_i\; \backslash frac\{x\_\{i+1\}-x\_i\}\{2\}(y\_i\; +\; y\_\{i+1\})$$

Numerical methods / root finding

Root of a polynomial by Newton's method

$$x_{k+1}=x_k-\frac{p(x_k)}{p^{\mathrm{\prime }}(x_k)}x\_\{k+1\}\; =\; x\_k\; -\; \backslash frac\{p(x\_k)\}\{p\text{'}(x\_k)\}$$

Root of a polynomial by bisection

Root of a polynomial by the secant method

$$x_{k+1}=x_k-p(x_k)\frac{x_k-x_{k-1}}{p(x_k)-p(x_{k-1})}x\_\{k+1\}\; =\; x\_k\; -\; p(x\_k)\backslash ,\backslash frac\{x\_k\; -\; x\_\{k-1\}\}\{p(x\_k)\; -\; p(x\_\{k-1\})\}$$

Numerical methods / splines

Clamped cubic spline interpolation

$$2h_0{M}_0+h_0{M}_1=6(\frac{y_1-y_0}{h_0}-s_0),h_{n-1}{M}_{n-1}+2h_{n-1}{M}_n=6(s_n-\frac{y_n-y_{n-1}}{h_{n-1}})2h\_0M\_0\; +\; h\_0M\_1\; =\; 6\backslash Big(\backslash frac\{y\_1-y\_0\}\{h\_0\}\; -\; s\_0\backslash Big),\backslash \; h\_\{n-1\}M\_\{n-1\}\; +\; 2h\_\{n-1\}M\_n\; =\; 6\backslash Big(s\_n\; -\; \backslash frac\{y\_n-y\_\{n-1\}\}\{h\_\{n-1\}\}\backslash Big)$$

Natural cubic spline interpolation

$$h_{i-1}{M}_{i-1}+2(h_{i-1}+h_i)M_i+h_i{M}_{i+1}=6(\frac{y_{i+1}-y_i}{h_i}-\frac{y_i-y_{i-1}}{h_{i-1}}),M_0=M_n=0h\_\{i-1\}M\_\{i-1\}\; +\; 2(h\_\{i-1\}+h\_i)M\_i\; +\; h\_iM\_\{i+1\}\; =\; 6\backslash Big(\backslash frac\{y\_\{i+1\}-y\_i\}\{h\_i\}\; -\; \backslash frac\{y\_i-y\_\{i-1\}\}\{h\_\{i-1\}\}\backslash Big),\backslash \; M\_0\; =\; M\_n\; =\; 0$$

Sparse matrices / sparse operations

Sparse matrix times vector (CSR)

$$y_r=\sum _{k=\mathtt{\text{indptr}}[r]}^{\mathtt{\text{indptr}}[r+1]-1}\mathtt{\text{data}}[k]x_{\mathtt{\text{indices}}[k]}y\_r\; =\; \backslash sum\_\{k=\backslash texttt\{indptr\}[r]\}^\{\backslash texttt\{indptr\}[r+1]-1\}\; \backslash texttt\{data\}[k]\backslash ,x\_\{\backslash texttt\{indices\}[k]\}$$

Sparse matrices / storage formats

Compressed sparse column (CSC) form of a matrix

$$\mathtt{\text{data}}[\mathtt{\text{indptr}}[k]:\mathtt{\text{indptr}}[k+1]]\backslash texttt\{data\}[\backslash texttt\{indptr\}[k]:\backslash texttt\{indptr\}[k+1]]$$

Compressed sparse row (CSR) form of a matrix

$$\mathtt{\text{data}}[\mathtt{\text{indptr}}[k]:\mathtt{\text{indptr}}[k+1]]\backslash texttt\{data\}[\backslash texttt\{indptr\}[k]:\backslash texttt\{indptr\}[k+1]]$$

Dense matrix from CSR arrays

Vector spaces / bases and coordinates

Coordinates of a vector in a basis

$$\mathbf{v}=\sum _i{c}_i{\mathbf{b}}_i\backslash mathbf\{v\}\; =\; \backslash sum\_i\; c\_i\; \backslash mathbf\{b\}\_i$$

Vector spaces / fundamental subspaces

Column space of a matrix

$$\mathrm{col}A=\mathrm{span}\{{\mathbf{a}}_j:j\text{ a pivot column}\}\backslash operatorname\{col\}A\; =\; \backslash operatorname\{span\}\backslash \{\backslash mathbf\{a\}\_j\; :\; j\; \backslash text\{\; a\; pivot\; column\}\backslash \}$$

Null space (kernel) of a matrix

$$\mathrm{null}A=\{\mathbf{x}:A\mathbf{x}=\mathbf{0}\},\dim =n-\mathrm{rank}A\backslash operatorname\{null\}A\; =\; \backslash \{\backslash mathbf\{x\}\; :\; A\backslash mathbf\{x\}\; =\; \backslash mathbf\{0\}\backslash \},\backslash quad\; \backslash dim\; =\; n\; -\; \backslash operatorname\{rank\}A$$

Vector spaces / independence and span

Linear independence of vectors

$$c_1{\mathbf{v}}_1+\cdots +c_k{\mathbf{v}}_k=\mathbf{0}\Rightarrow c_1=\cdots =c_k=0c\_1\backslash mathbf\{v\}\_1\; +\; \backslash cdots\; +\; c\_k\backslash mathbf\{v\}\_k\; =\; \backslash mathbf\{0\}\; \backslash Rightarrow\; c\_1\; =\; \backslash cdots\; =\; c\_k\; =\; 0$$

Vector spaces / orthogonality

Gram-Schmidt orthonormalization

Vectors / length and distance

Distance between two points

$$d=\sqrt{{\sum }_i(p_i-r_i{)}^2}d\; =\; \backslash sqrt\{\backslash textstyle\backslash sum\_i\; (p\_i\; -\; r\_i)^2\}$$

Length (norm) of a vector

$$vert\mathbf{v}ver{t}_2=\sqrt{{\sum }_i{v}_i^2}\backslash \backslash vert\; \backslash mathbf\{v\}\backslash \backslash vert\; \_2\; =\; \backslash sqrt\{\backslash textstyle\backslash sum\_i\; v\_i^2\}$$

Unit vector in the direction of a vector

$$\widehat{\mathbf{v}}=\mathbf{v}\mathrm{/}vert\mathbf{v}vert\backslash hat\{\backslash mathbf\{v\}\}\; =\; \backslash mathbf\{v\}/\backslash \backslash vert\; \backslash mathbf\{v\}\backslash \backslash vert$$

Vectors / products of vectors

Angle between two vectors

$$\cos \theta =\frac{\mathbf{u}\cdot \mathbf{v}}{vert\mathbf{u}vertvert\mathbf{v}vert}\backslash cos\backslash theta\; =\; \backslash frac\{\backslash mathbf\{u\}\backslash cdot\backslash mathbf\{v\}\}\{\backslash \backslash vert\; \backslash mathbf\{u\}\backslash \backslash vert\; \backslash ,\backslash \backslash vert\; \backslash mathbf\{v\}\backslash \backslash vert\; \}$$

Area of the parallelogram (and triangle) spanned by two vectors

$$A=vert\mathbf{u}\times \mathbf{v}vert=\sqrt{vert\mathbf{u}ver{t}^{2}vert\mathbf{v}ver{t}^2-(\mathbf{u}\cdot \mathbf{v}{)}^2}A\; =\; \backslash \backslash vert\; \backslash mathbf\{u\}\backslash times\backslash mathbf\{v\}\backslash \backslash vert\; =\; \backslash sqrt\{\backslash \backslash vert\; \backslash mathbf\{u\}\backslash \backslash vert\; ^2\backslash \backslash vert\; \backslash mathbf\{v\}\backslash \backslash vert\; ^2\; -\; (\backslash mathbf\{u\}\backslash cdot\backslash mathbf\{v\})^2\}$$

Cross product of two 3-D vectors

$$\mathbf{u}\times \mathbf{v}=(u_2{v}_3-u_3{v}_2,u_3{v}_1-u_1{v}_3,u_1{v}_2-u_2{v}_1)\backslash mathbf\{u\}\backslash times\backslash mathbf\{v\}\; =\; (u\_2v\_3-u\_3v\_2,\backslash ;\; u\_3v\_1-u\_1v\_3,\backslash ;\; u\_1v\_2-u\_2v\_1)$$

Dot product of two vectors

$$\mathbf{u}\cdot \mathbf{v}=\sum _i{u}_i{v}_i\backslash mathbf\{u\}\backslash cdot\backslash mathbf\{v\}\; =\; \backslash sum\_i\; u\_i\; v\_i$$

Outer product of two vectors

$$(\mathbf{u}{\mathbf{v}}^T{)}_{ij}=u_i{v}_j(\backslash mathbf\{u\}\backslash mathbf\{v\}^\{T\})\_\{ij\}\; =\; u\_i\; v\_j$$

Scalar triple product (volume of a parallelepiped)

$$\mathbf{u}\cdot (\mathbf{v}\times \mathbf{w})=\det [\mathbf{u};\mathbf{v};\mathbf{w}]\backslash mathbf\{u\}\backslash cdot(\backslash mathbf\{v\}\backslash times\backslash mathbf\{w\})\; =\; \backslash det[\backslash mathbf\{u\};\backslash mathbf\{v\};\backslash mathbf\{w\}]$$

Vectors / projections

Projection of one vector onto another

$${\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}}_{\mathbf{v}}\mathbf{u}=\frac{\mathbf{u}\cdot \mathbf{v}}{\mathbf{v}\cdot \mathbf{v}}\mathbf{v}\backslash mathrm\{proj\}\_\{\backslash mathbf\{v\}\}\backslash mathbf\{u\}\; =\; \backslash frac\{\backslash mathbf\{u\}\backslash cdot\backslash mathbf\{v\}\}\{\backslash mathbf\{v\}\backslash cdot\backslash mathbf\{v\}\}\backslash ,\backslash mathbf\{v\}$$

Vectors / vector arithmetic

Difference of two vectors

$$\mathbf{u}-\mathbf{v}=(u_1-v_1,\dots ,u_n-v_n)\backslash mathbf\{u\}-\backslash mathbf\{v\}\; =\; (u\_1-v\_1,\; \backslash ldots,\; u\_n-v\_n)$$

Scalar multiple of a vector

$$k\mathbf{v}=(k{v}_1,\dots ,k{v}_n)k\backslash mathbf\{v\}\; =\; (k\; v\_1,\; \backslash ldots,\; k\; v\_n)$$

Sum of two vectors

$$\mathbf{u}+\mathbf{v}=(u_1+v_1,\dots ,u_n+v_n)\backslash mathbf\{u\}+\backslash mathbf\{v\}\; =\; (u\_1+v\_1,\; \backslash ldots,\; u\_n+v\_n)$$

Out of scope

• Open-ended explanation, essay writing or tutoring dialogue: it is a solver, not a chatbot.
• Questions outside its certified linear algebra capabilities, and any question whose stated numbers it cannot all use — it refuses with the reason instead.
• High-stakes engineering, medical or legal decisions without independent review.

Limitations

• It answers what its certified capabilities cover (vectors, matrices, determinants, linear systems, eigenvalues, decompositions, least squares, sparse matrices and numerical methods (root finding, interpolation, quadrature)); anything else is refused with a reason.
• ask() reads documented question forms and common phrasings; unusual wording may be refused — use solve() for guaranteed results.
• Results are as correct as the certified relations and their stated conditions; a certified relation is a checked model, not a claim about every physical situation.

Citation

@software{linear_algebra_ai999x_1_0_3,
  title  = {linear-algebra-ai999x: an offline, verified linear algebra solver},
  author = {Farabi, Ibnul Jaif}, organization = {AI999X},
  year   = {2026}, version = {1.0.3},
  url    = {https://huggingface.co/AI999X/linear-algebra-ai999x}
}

Contact

Questions, bug reports and requests: open a discussion on https://huggingface.co/AI999X/linear-algebra-ai999x/discussions or visit https://ai999x.com.

Support the project

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Created and engineered by Ibnul Jaif Farabi · AI999X · ai999x.com