Linear Algebra Visualization
Matrix transformation visualizations, eigenvector geometry, SVD diagram interpretation.
Subject Overview
20%
of Total Question Bank
4th largest discipline
6
Sub-Domains
Algebra·Geometry·Calculus·Probability·Discrete·Numerical
50%
Diagram-Reasoning
Graph·Matrix·Venn·Coordinate·Network·Tree
Mathematics uniquely demands the fusion of symbolic reasoning (LaTeX formulas, algebraic derivations) with visual reasoning (graphs, geometric figures, network diagrams). This dual-channel difficulty sets math apart: models that excel at text-only math benchmarks (MATH, GSM8K) often fail when the same problem is presented as a diagram or requires interpretation of a coordinate graph alongside LaTeX. Langhui Mathematics benchmark systematically tests this gap: 50% of questions require diagram + LaTeX co-reasoning.
Sub-Domains & High-Value Difficulty Zones
Algebra (25%)
Abstract Algebra
Group theory Cayley tables, ring/field lattice diagrams, Galois correspondence visualizations.
Number Theory
Modular arithmetic visualizations, lattice point diagrams, RSA encryption schemas.
Geometry (25%)
Plane & Solid Geometry
Angle/chord theorems, inscribed/circumscribed figures, 3D cross-section reasoning.
Analytic Geometry
Coordinate geometry with graphs, conic section identification, parametric curve tracing.
Topology
Homeomorphism diagrams, knot theory, fundamental group visualizations.
Evidence: Topology diagram reasoning is near-zero for open-source VLMs.
Calculus (20%)
Differentiation & Integration
Function graph interpretation, area under curve, Riemann sum visualization.
Multivariable Calculus
3D surface plots, gradient vector fields, double/triple integral regions.
Differential Equations
Slope fields, phase portraits, stability analysis diagrams.
Probability & Statistics (15%)
Probability Diagrams
Bayes theorem via Venn diagrams, probability trees, Markov chain state diagrams.
Evidence: 先进 models confuse conditional vs joint probability in Venn contexts.
Statistical Inference
Confidence interval visualizations, hypothesis test diagrams, QQ/PP plots.
Discrete Mathematics (10%)
Graph Theory
Shortest path, MST, Eulerian/Hamiltonian path diagrams, bipartite matching.
Combinatorics
Pascal's triangle, inclusion-exclusion via set diagrams, partition lattices.
Numerical Analysis (5%)
Numerical Methods Visualization
Newton-Raphson iteration diagrams, convergence rate plots, interpolation error graphs.
Optimization
Gradient descent trajectories, Lagrange multiplier geometric interpretation.
Difficulty Distribution
| Level | Stage | Share | Typical Question Types |
|---|---|---|---|
| L1 Lower Undergrad | Calculus I-II, Linear Algebra | 15% | Single-variable calculus with graphs, basic matrix operations |
| L2 Upper Undergrad | Advanced calculus, abstract algebra, probability | 50% | Multivariable calculus with 3D plots, eigenvalue analysis, Bayes reasoning |
| L3 Graduate Entry | GRE Math / Qualifying exams | 25% | Proof-based questions, topology, advanced graph theory |
| L4 Research | PhD / Putnam competition | 10% | Open-ended proofs, novel problem construction |
Sample Questions
STEM-2026-MAT-0000234L2 Upper Undergrad
Linear Algebra · Matrix Transformation — Diagram Reasoning
Prompt: A 2×2 matrix A = [[2,1],[1,2]] is applied to the unit square in R². The accompanying diagram shows the original unit square and its image after transformation. Identify the geometric interpretation: (a) rotation, (b) reflection, (c) shear, (d) scaling with shear. Additionally, diagonalize A and interpret the eigenvalues geometrically on the diagram.
Answer: The transformation is scaling with shear (d). Diagonalization: λ1=3, v1=(1,1); λ2=1, v2=(-1,1). Eigenvalue 3 stretches along (1,1), eigenvalue 1 preserves along (-1,1). The image parallelogram shows the unit square stretched 3× along the (1,1) direction.
Step-by-Step Solution
- Compute det(A-λI) = (2-λ)²-1 = λ²-4λ+3 = (λ-3)(λ-1).
- For λ1=3: (A-3I)v=0 → [[-1,1],[1,-1]]v=0 → v1=(1,1).
- For λ2=1: (A-I)v=0 → [[1,1],[1,1]]v=0 → v2=(-1,1).
- Geometric interpretation: unit square transforms to parallelogram with vertices (0,0), (2,1), (1,2), (3,3). The (1,1) direction is stretched by factor 3.
- This is a symmetric positive definite matrix, so it's pure scaling along orthogonal eigenvectors — no rotation, no reflection.
5-Model Evaluation
GPT-5.1
2/5
Claude Opus 4.6
1/5
Gemini-3.1-Pro
2/5
Qwen3.6-Plus
1/5
DeepSeek-V4
0/5
STEM-2026-MAT-0000235L2 Upper Undergrad
Probability · Bayes' Theorem with Venn Diagram
Prompt: A Venn diagram shows two overlapping events A and B within sample space S. Given P(A)=0.3, P(B)=0.4, P(A∩B)=0.2, calculate P(A|B) and P(B|A). Additionally, shade the region representing P(A|B) on the diagram. Which is larger and why?
Answer: P(A|B) = P(A∩B)/P(B) = 0.2/0.4 = 0.5. P(B|A) = P(A∩B)/P(A) = 0.2/0.3 ≈ 0.667. P(B|A) > P(A|B) because event B accounts for a larger proportion of A (67%) than A accounts for B (50%). The shaded region for P(A|B) is the intersection A∩B normalized by the total area of B.
5-Model Evaluation
GPT-5.1
3/5
Claude Opus 4.6
2/5
Gemini-3.1-Pro
2/5
Qwen3.6-Plus
1/5
DeepSeek-V4
1/5
STEM-2026-MAT-0000236L3 Graduate Entry
Discrete Math · Dijkstra's Shortest Path
Prompt: A weighted directed graph with 6 nodes is shown. Edge weights are labeled. Apply Dijkstra's algorithm from node A to find the shortest path to node F. List the order of node relaxation and the final shortest path with total cost. Then identify the worst-case time complexity if the graph were dense (|E|≈|V|²).
Answer: Shortest path: A→C→D→F with total cost 14. Relaxation order: A(0), C(6), B(8), D(11), E(15), F(14). With adjacency list + binary heap: O((V+E)logV). With dense graph and Fibonacci heap: O(V²·logV) using adjacency matrix, or O(E+V·logV) ≈ O(V²) with binary heap for dense graphs.
Step-by-Step Reasoning
- Initialize: dist[A]=0, all others ∞. Unvisited: {A,B,C,D,E,F}.
- Visit A: relax edges A→B(8), A→C(6). dist[B]=8, dist[C]=6.
- Visit C (min dist=6): relax C→D(5), C→E(10). dist[D]=11, dist[E]=16.
- Visit B (min dist=8): relax B→D(2). dist[D]=min(11,10)=10 via B→D, but A→C→D=11 still. Check: B→D(2)→F(7)=17 > 14.
- Visit D (min dist=10): relax D→F(4). dist[F]=14.
- Visit F: done, no outgoing edges to unvisited nodes.
- Final path reconstruction: F←D←C←A. Cost: 6+5+4=14.
5-Model Evaluation
GPT-5.1
2/5
Claude Opus 4.6
1/5
Gemini-3.1-Pro
2/5
Qwen3.6-Plus
0/5
DeepSeek-V4
1/5
Delivery & API
JSONL Bulk Download: Full fields + image URLs + LaTeX source + 5-model evaluation + reasoning chains.
REST API: GET /DataAssetsAPI/stem-questions/mathematics. For enterprise API keys, contact lk@langhuiai.com.
FAQ
How is the difficulty threshold enforced?
Every question must achieve ≤40% pass rate across 5 models × 5 passes (GPT-5.1 / Claude Opus 4.6 / Gemini-3.1-Pro / Qwen3.6-Plus / DeepSeek-V4). Human expert accuracy ≥95%, dual-review.
What sub-domains does the Mathematics benchmark cover?
6 sub-domains: Algebra (25%), Geometry (25%), Calculus (20%), Probability & Statistics (15%), Discrete Mathematics (10%), Numerical Analysis (5%).
Why 50% diagram-reasoning questions?
Models that excel at text-only math (MATH, GSM8K) often fail when the same problem requires interpreting a diagram alongside LaTeX. We specifically design 50% of questions with visual math components: graphs, geometric figures, network diagrams, Venn diagrams, 3D surface plots, matrix transformations.
Are LaTeX formulas included in the dataset?
Yes. All questions include both a rendered image of LaTeX formulas (for visual recognition) and the raw LaTeX source code (for text-based training). Models must bridge visual formula rendering and symbolic manipulation.
Is there overlap with public math benchmarks?
Zero overlap with MATH, GSM8K, MathVista, MMMU-Math, and 8 other public benchmarks. <0.5% pHash+n-gram fingerprint overlap.