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queries/queries_linear-algebra.jsonl

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+{"id": "linear-algebra_T1_58", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Row Scaling?", "ground_truth": ["Row Scaling", "LINSYS"], "concept_id": 58, "hop_depth": 0}
+{"id": "linear-algebra_T1_13", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Unit Vector?", "ground_truth": ["Unit Vector", "FOUND"], "concept_id": 13, "hop_depth": 0}
+{"id": "linear-algebra_T1_141", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Left Singular Vector?", "ground_truth": ["Left Singular Vector", "DECOMP"], "concept_id": 141, "hop_depth": 0}
+{"id": "linear-algebra_T1_126", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Spectral Theorem?", "ground_truth": ["Spectral Theorem", "EIGEN"], "concept_id": 126, "hop_depth": 0}
+{"id": "linear-algebra_T1_115", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Eigenvector?", "ground_truth": ["Eigenvector", "EIGEN"], "concept_id": 115, "hop_depth": 0}
+{"id": "linear-algebra_T1_72", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Numerical Stability?", "ground_truth": ["Numerical Stability", "LINSYS"], "concept_id": 72, "hop_depth": 0}
+{"id": "linear-algebra_T1_53", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Matrix Equation Form?", "ground_truth": ["Matrix Equation Form", "LINSYS"], "concept_id": 53, "hop_depth": 0}
+{"id": "linear-algebra_T1_280", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Point Cloud?", "ground_truth": ["Point Cloud", "GEOM3D"], "concept_id": 280, "hop_depth": 0}
+{"id": "linear-algebra_T1_45", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Orthogonal Matrix?", "ground_truth": ["Orthogonal Matrix", "MATOP"], "concept_id": 45, "hop_depth": 0}
+{"id": "linear-algebra_T1_217", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Word Embedding?", "ground_truth": ["Word Embedding", "GENAI"], "concept_id": 217, "hop_depth": 0}
+{"id": "linear-algebra_T1_17", "domain": "linear-algebra", "type": "T1_entity", "query": "What is L2 Norm?", "ground_truth": ["L2 Norm", "FOUND"], "concept_id": 17, "hop_depth": 0}
+{"id": "linear-algebra_T1_16", "domain": "linear-algebra", "type": "T1_entity", "query": "What is L1 Norm?", "ground_truth": ["L1 Norm", "FOUND"], "concept_id": 16, "hop_depth": 0}
+{"id": "linear-algebra_T1_48", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Sparse Matrix?", "ground_truth": ["Sparse Matrix", "MATOP"], "concept_id": 48, "hop_depth": 0}
+{"id": "linear-algebra_T1_112", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Signed Area?", "ground_truth": ["Signed Area", "DETERM"], "concept_id": 112, "hop_depth": 0}
+{"id": "linear-algebra_T1_120", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Algebraic Multiplicity?", "ground_truth": ["Algebraic Multiplicity", "EIGEN"], "concept_id": 120, "hop_depth": 0}
+{"id": "linear-algebra_T1_259", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Frequency Domain?", "ground_truth": ["Frequency Domain", "IMGPROC"], "concept_id": 259, "hop_depth": 0}
+{"id": "linear-algebra_T1_14", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Vector Normalization?", "ground_truth": ["Vector Normalization", "FOUND"], "concept_id": 14, "hop_depth": 0}
+{"id": "linear-algebra_T1_288", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Update Step?", "ground_truth": ["Update Step", "AUTON"], "concept_id": 288, "hop_depth": 0}
+{"id": "linear-algebra_T1_102", "domain": "linear-algebra", "type": "T1_entity", "query": "What is 2x2 Determinant?", "ground_truth": ["2x2 Determinant", "DETERM"], "concept_id": 102, "hop_depth": 0}
+{"id": "linear-algebra_T1_215", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Embedding?", "ground_truth": ["Embedding", "GENAI"], "concept_id": 215, "hop_depth": 0}
+{"id": "linear-algebra_T1_113", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Cramers Rule?", "ground_truth": ["Cramers Rule", "DETERM"], "concept_id": 113, "hop_depth": 0}
+{"id": "linear-algebra_T1_230", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Position Encoding?", "ground_truth": ["Position Encoding", "GENAI"], "concept_id": 230, "hop_depth": 0}
+{"id": "linear-algebra_T1_143", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Full SVD?", "ground_truth": ["Full SVD", "DECOMP"], "concept_id": 143, "hop_depth": 0}
+{"id": "linear-algebra_T1_4", "domain": "linear-algebra", "type": "T1_entity", "query": "What is 2D Vector?", "ground_truth": ["2D Vector", "FOUND"], "concept_id": 4, "hop_depth": 0}
+{"id": "linear-algebra_T1_82", "domain": "linear-algebra", "type": "T1_entity", "query": "What is 3D Rotation?", "ground_truth": ["3D Rotation", "TRANS"], "concept_id": 82, "hop_depth": 0}
+{"id": "linear-algebra_T1_175", "domain": "linear-algebra", "type": "T1_entity", "query": "What is PCA?", "ground_truth": ["PCA", "MLBASE"], "concept_id": 175, "hop_depth": 0}
+{"id": "linear-algebra_T1_80", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Rotation Matrix?", "ground_truth": ["Rotation Matrix", "TRANS"], "concept_id": 80, "hop_depth": 0}
+{"id": "linear-algebra_T1_111", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Volume Scaling Factor?", "ground_truth": ["Volume Scaling Factor", "DETERM"], "concept_id": 111, "hop_depth": 0}
+{"id": "linear-algebra_T1_173", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Correlation Matrix?", "ground_truth": ["Correlation Matrix", "MLBASE"], "concept_id": 173, "hop_depth": 0}
+{"id": "linear-algebra_T1_195", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Softmax?", "ground_truth": ["Softmax", "NEURAL"], "concept_id": 195, "hop_depth": 0}
+{"id": "linear-algebra_T1_50", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Block Matrix?", "ground_truth": ["Block Matrix", "MATOP"], "concept_id": 50, "hop_depth": 0}
+{"id": "linear-algebra_T1_184", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Regularization?", "ground_truth": ["Regularization", "MLBASE"], "concept_id": 184, "hop_depth": 0}
+{"id": "linear-algebra_T1_177", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Variance Explained?", "ground_truth": ["Variance Explained", "MLBASE"], "concept_id": 177, "hop_depth": 0}
+{"id": "linear-algebra_T1_136", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Householder QR?", "ground_truth": ["Householder QR", "DECOMP"], "concept_id": 136, "hop_depth": 0}
+{"id": "linear-algebra_T1_23", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Basis Vector?", "ground_truth": ["Basis Vector", "FOUND"], "concept_id": 23, "hop_depth": 0}
+{"id": "linear-algebra_T1_236", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Convex Function?", "ground_truth": ["Convex Function", "OPTIM"], "concept_id": 236, "hop_depth": 0}
+{"id": "linear-algebra_T1_275", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Projection Matrix?", "ground_truth": ["Projection Matrix", "GEOM3D"], "concept_id": 275, "hop_depth": 0}
+{"id": "linear-algebra_T1_64", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Free Variable?", "ground_truth": ["Free Variable", "LINSYS"], "concept_id": 64, "hop_depth": 0}
+{"id": "linear-algebra_T1_194", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Tanh?", "ground_truth": ["Tanh", "NEURAL"], "concept_id": 194, "hop_depth": 0}
+{"id": "linear-algebra_T1_41", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Diagonal Matrix?", "ground_truth": ["Diagonal Matrix", "MATOP"], "concept_id": 41, "hop_depth": 0}
+{"id": "linear-algebra_T1_283", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Sensor Fusion?", "ground_truth": ["Sensor Fusion", "AUTON"], "concept_id": 283, "hop_depth": 0}
+{"id": "linear-algebra_T1_151", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Subspace?", "ground_truth": ["Subspace", "INPROD"], "concept_id": 151, "hop_depth": 0}
+{"id": "linear-algebra_T1_186", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Gradient Descent?", "ground_truth": ["Gradient Descent", "MLBASE"], "concept_id": 186, "hop_depth": 0}
+{"id": "linear-algebra_T1_296", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Object Tracking?", "ground_truth": ["Object Tracking", "AUTON"], "concept_id": 296, "hop_depth": 0}
+{"id": "linear-algebra_T1_99", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Change of Basis?", "ground_truth": ["Change of Basis", "TRANS"], "concept_id": 99, "hop_depth": 0}
+{"id": "linear-algebra_T1_36", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Matrix-Vector Product?", "ground_truth": ["Matrix-Vector Product", "MATOP"], "concept_id": 36, "hop_depth": 0}
+{"id": "linear-algebra_T1_24", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Standard Basis?", "ground_truth": ["Standard Basis", "FOUND"], "concept_id": 24, "hop_depth": 0}
+{"id": "linear-algebra_T1_117", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Characteristic Polynomial?", "ground_truth": ["Characteristic Polynomial", "EIGEN"], "concept_id": 117, "hop_depth": 0}
+{"id": "linear-algebra_T1_149", "domain": "linear-algebra", "type": "T1_entity", "query": "What is Condition Number?", "ground_truth": ["Condition Number", "DECOMP"], "concept_id": 149, "hop_depth": 0}
+{"id": "linear-algebra_T1_52", "domain": "linear-algebra", "type": "T1_entity", "query": "What is System of Equations?", "ground_truth": ["System of Equations", "LINSYS"], "concept_id": 52, "hop_depth": 0}
+{"id": "linear-algebra_T2_197", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Bias Vector?", "ground_truth": ["Vector", "Neuron Model"], "concept_id": 197, "hop_depth": 1}
+{"id": "linear-algebra_T2_145", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Truncated SVD?", "ground_truth": ["Compact SVD"], "concept_id": 145, "hop_depth": 1}
+{"id": "linear-algebra_T2_235", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Convexity?", "ground_truth": ["Hessian Matrix"], "concept_id": 235, "hop_depth": 1}
+{"id": "linear-algebra_T2_189", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Perceptron?", "ground_truth": ["Dot Product", "Neuron Model"], "concept_id": 189, "hop_depth": 1}
+{"id": "linear-algebra_T2_86", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Shear Matrix?", "ground_truth": ["Transformation Matrix"], "concept_id": 86, "hop_depth": 1}
+{"id": "linear-algebra_T2_192", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for ReLU?", "ground_truth": ["Activation Function"], "concept_id": 192, "hop_depth": 1}
+{"id": "linear-algebra_T2_184", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Regularization?", "ground_truth": ["Linear Regression"], "concept_id": 184, "hop_depth": 1}
+{"id": "linear-algebra_T2_110", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Singular Matrix?", "ground_truth": ["Invertible Matrix", "Determinant"], "concept_id": 110, "hop_depth": 1}
+{"id": "linear-algebra_T2_139", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for SVD?", "ground_truth": ["Matrix Factorization", "QR Decomposition"], "concept_id": 139, "hop_depth": 1}
+{"id": "linear-algebra_T2_38", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Matrix Transpose?", "ground_truth": ["Matrix", "Row Vector", "Column Vector"], "concept_id": 38, "hop_depth": 1}
+{"id": "linear-algebra_T2_90", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Composition of Transforms?", "ground_truth": ["Linear Transformation", "Matrix Multiplication"], "concept_id": 90, "hop_depth": 1}
+{"id": "linear-algebra_T2_276", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Perspective Projection?", "ground_truth": ["Projection Matrix"], "concept_id": 276, "hop_depth": 1}
+{"id": "linear-algebra_T2_128", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Power Iteration?", "ground_truth": ["Eigenvector", "Matrix Multiplication"], "concept_id": 128, "hop_depth": 1}
+{"id": "linear-algebra_T2_239", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for BFGS Algorithm?", "ground_truth": ["Quasi-Newton Method"], "concept_id": 239, "hop_depth": 1}
+{"id": "linear-algebra_T2_141", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Left Singular Vector?", "ground_truth": ["SVD", "Singular Value"], "concept_id": 141, "hop_depth": 1}
+{"id": "linear-algebra_T2_288", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Update Step?", "ground_truth": ["Kalman Filter", "Prediction Step"], "concept_id": 288, "hop_depth": 1}
+{"id": "linear-algebra_T2_115", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Eigenvector?", "ground_truth": ["Eigenvalue"], "concept_id": 115, "hop_depth": 1}
+{"id": "linear-algebra_T2_169", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Feature Vector?", "ground_truth": ["N-Dimensional Vector", "Data Matrix"], "concept_id": 169, "hop_depth": 1}
+{"id": "linear-algebra_T2_30", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Matrix Dimensions?", "ground_truth": ["Matrix", "Matrix Notation"], "concept_id": 30, "hop_depth": 1}
+{"id": "linear-algebra_T2_120", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Algebraic Multiplicity?", "ground_truth": ["Characteristic Polynomial", "Characteristic Equation"], "concept_id": 120, "hop_depth": 1}
+{"id": "linear-algebra_T2_18", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for L-Infinity Norm?", "ground_truth": ["Vector", "Vector Magnitude"], "concept_id": 18, "hop_depth": 1}
+{"id": "linear-algebra_T2_164", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Normal Equations?", "ground_truth": ["Matrix Transpose", "Least Squares Problem"], "concept_id": 164, "hop_depth": 1}
+{"id": "linear-algebra_T2_208", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Stride?", "ground_truth": ["Convolutional Layer", "Convolution Kernel"], "concept_id": 208, "hop_depth": 1}
+{"id": "linear-algebra_T2_140", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Singular Value?", "ground_truth": ["SVD"], "concept_id": 140, "hop_depth": 1}
+{"id": "linear-algebra_T2_35", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Matrix Scalar Multiply?", "ground_truth": ["Scalar", "Matrix"], "concept_id": 35, "hop_depth": 1}
+{"id": "linear-algebra_T2_111", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Volume Scaling Factor?", "ground_truth": ["Determinant", "3x3 Determinant"], "concept_id": 111, "hop_depth": 1}
+{"id": "linear-algebra_T2_293", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Localization?", "ground_truth": ["State Estimation"], "concept_id": 293, "hop_depth": 1}
+{"id": "linear-algebra_T2_258", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Fourier Transform?", "ground_truth": ["Matrix", "Matrix Multiplication"], "concept_id": 258, "hop_depth": 1}
+{"id": "linear-algebra_T2_205", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Deep Network?", "ground_truth": ["Hidden Layer"], "concept_id": 205, "hop_depth": 1}
+{"id": "linear-algebra_T2_237", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Newtons Method?", "ground_truth": ["Gradient Descent", "Hessian Matrix"], "concept_id": 237, "hop_depth": 1}
+{"id": "linear-algebra_T2_76", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Transformation Matrix?", "ground_truth": ["Matrix Multiplication", "Linear Transformation"], "concept_id": 76, "hop_depth": 1}
+{"id": "linear-algebra_T2_138", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Positive Definite Matrix?", "ground_truth": ["Symmetric Matrix", "Eigenvalue"], "concept_id": 138, "hop_depth": 1}
+{"id": "linear-algebra_T2_73", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Back Substitution?", "ground_truth": ["Row Echelon Form", "Upper Triangular"], "concept_id": 73, "hop_depth": 1}
+{"id": "linear-algebra_T2_129", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Dominant Eigenvalue?", "ground_truth": ["Power Iteration"], "concept_id": 129, "hop_depth": 1}
+{"id": "linear-algebra_T2_290", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Extended Kalman Filter?", "ground_truth": ["Kalman Filter", "Newtons Method"], "concept_id": 290, "hop_depth": 1}
+{"id": "linear-algebra_T2_278", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Triangulation?", "ground_truth": ["3D Vector", "Stereo Vision"], "concept_id": 278, "hop_depth": 1}
+{"id": "linear-algebra_T2_137", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Cholesky Decomposition?", "ground_truth": ["Lower Triangular", "Matrix Factorization", "Positive Definite Matrix"], "concept_id": 137, "hop_depth": 1}
+{"id": "linear-algebra_T2_222", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Cross-Attention?", "ground_truth": ["Attention Mechanism", "Self-Attention"], "concept_id": 222, "hop_depth": 1}
+{"id": "linear-algebra_T2_207", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Convolution Kernel?", "ground_truth": ["Convolutional Layer"], "concept_id": 207, "hop_depth": 1}
+{"id": "linear-algebra_T2_188", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Learning Rate?", "ground_truth": ["Gradient Descent"], "concept_id": 188, "hop_depth": 1}
+{"id": "linear-algebra_T2_72", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Numerical Stability?", "ground_truth": ["Gaussian Elimination", "Row Echelon Form"], "concept_id": 72, "hop_depth": 1}
+{"id": "linear-algebra_T2_263", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Homography?", "ground_truth": ["Transformation Matrix", "Homogeneous Coordinates"], "concept_id": 263, "hop_depth": 1}
+{"id": "linear-algebra_T2_255", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Sharpen Filter?", "ground_truth": ["Image Filter"], "concept_id": 255, "hop_depth": 1}
+{"id": "linear-algebra_T2_48", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Sparse Matrix?", "ground_truth": ["Matrix", "Matrix Entry"], "concept_id": 48, "hop_depth": 1}
+{"id": "linear-algebra_T2_26", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Vector Space?", "ground_truth": ["Linear Combination", "Linear Independence", "Basis Vector"], "concept_id": 26, "hop_depth": 1}
+{"id": "linear-algebra_T2_58", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Row Scaling?", "ground_truth": ["Row Operations"], "concept_id": 58, "hop_depth": 1}
+{"id": "linear-algebra_T2_81", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for 2D Rotation?", "ground_truth": ["2D Vector", "Rotation Matrix"], "concept_id": 81, "hop_depth": 1}
+{"id": "linear-algebra_T2_84", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Uniform Scaling?", "ground_truth": ["Scaling Matrix"], "concept_id": 84, "hop_depth": 1}
+{"id": "linear-algebra_T2_219", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Cosine Similarity?", "ground_truth": ["Unit Vector", "Semantic Similarity"], "concept_id": 219, "hop_depth": 1}
+{"id": "linear-algebra_T2_34", "domain": "linear-algebra", "type": "T2_dependency", "query": "What are the prerequisites for Matrix Addition?", "ground_truth": ["Matrix", "Matrix Entry"], "concept_id": 34, "hop_depth": 1}
+{"id": "linear-algebra_T3_74_167", "domain": "linear-algebra", "type": "T3_path", "query": "What is the prerequisite chain from Function to Four Subspaces?", "ground_truth": ["Four Subspaces", "Null Space", "Kernel", "Linear Transformation", "Function"], "concept_id": 167, "hop_depth": 4, "path_ids": [167, 92, 91, 75, 74]}
+{"id": "linear-algebra_T3_74_222", "domain": "linear-algebra", "type": "T3_path", "query": "What is the prerequisite chain from Function to Cross-Attention?", "ground_truth": ["Cross-Attention", "Attention Mechanism", "Weight Matrix", "Neuron Model", "Activation Function", "Function"], "concept_id": 222, "hop_depth": 5, "path_ids": [222, 220, 196, 190, 191, 74]}
+{"id": "linear-algebra_T3_1_271", "domain": "linear-algebra", "type": "T3_path", "query": "What is the prerequisite chain from Scalar to SE3 Transform?", "ground_truth": ["SE3 Transform", "Rigid Body Transform", "Homogeneous Coordinates", "N-Dimensional Vector", "Vector", "Scalar"], "concept_id": 271, "hop_depth": 5, "path_ids": [271, 270, 269, 6, 2, 1]}
+{"id": "linear-algebra_T3_1_271", "domain": "linear-algebra", "type": "T3_path", "query": "What is the prerequisite chain from Scalar to SE3 Transform?", "ground_truth": ["SE3 Transform", "Rigid Body Transform", "Homogeneous Coordinates", "N-Dimensional Vector", "Vector", "Scalar"], "concept_id": 271, "hop_depth": 5, "path_ids": [271, 270, 269, 6, 2, 1]}
+{"id": "linear-algebra_T3_74_261", "domain": "linear-algebra", "type": "T3_path", "query": "What is the prerequisite chain from Function to Color Space Transform?", "ground_truth": ["Color Space Transform", "Transformation Matrix", "Linear Transformation", "Function"], "concept_id": 261, "hop_depth": 3, "path_ids": [261, 76, 75, 74]}
+{"id": "linear-algebra_T3_74_189", "domain": "linear-algebra", "type": "T3_path", "query": "What is the prerequisite chain from Function to Perceptron?", "ground_truth": ["Perceptron", "Neuron Model", "Activation Function", "Function"], "concept_id": 189, "hop_depth": 3, "path_ids": [189, 190, 191, 74]}
+{"id": "linear-algebra_T3_1_200", "domain": "linear-algebra", "type": "T3_path", "query": "What is the prerequisite chain from Scalar to Chain Rule Matrices?", "ground_truth": ["Chain Rule Matrices", "Matrix Multiplication", "Matrix-Vector Product", "Matrix Scalar Multiply", "Scalar"], "concept_id": 200, "hop_depth": 4, "path_ids": [200, 37, 36, 35, 1]}
+{"id": "linear-algebra_T3_1_289", "domain": "linear-algebra", "type": "T3_path", "query": "What is the prerequisite chain from Scalar to Kalman Gain?", "ground_truth": ["Kalman Gain", "Update Step", "Kalman Filter", "State Vector", "Vector", "Scalar"], "concept_id": 289, "hop_depth": 5, "path_ids": [289, 288, 284, 285, 2, 1]}
+{"id": "linear-algebra_T3_1_214", "domain": "linear-algebra", "type": "T3_path", "query": "What is the prerequisite chain from Scalar to Tensor Operations?", "ground_truth": ["Tensor Operations", "Matrix Multiplication", "Matrix-Vector Product", "Matrix Scalar Multiply", "Scalar"], "concept_id": 214, "hop_depth": 4, "path_ids": [214, 37, 36, 35, 1]}
+{"id": "linear-algebra_T3_1_259", "domain": "linear-algebra", "type": "T3_path", "query": "What is the prerequisite chain from Scalar to Frequency Domain?", "ground_truth": ["Frequency Domain", "Fourier Transform", "Matrix", "Vector", "Scalar"], "concept_id": 259, "hop_depth": 4, "path_ids": [259, 258, 28, 2, 1]}
+{"id": "linear-algebra_T3_1_81", "domain": "linear-algebra", "type": "T3_path", "query": "What is the prerequisite chain from Scalar to 2D Rotation?", "ground_truth": ["2D Rotation", "2D Vector", "Vector", "Scalar"], "concept_id": 81, "hop_depth": 3, "path_ids": [81, 4, 2, 1]}
+{"id": "linear-algebra_T3_74_84", "domain": "linear-algebra", "type": "T3_path", "query": "What is the prerequisite chain from Function to Uniform Scaling?", "ground_truth": ["Uniform Scaling", "Scaling Matrix", "Transformation Matrix", "Linear Transformation", "Function"], "concept_id": 84, "hop_depth": 4, "path_ids": [84, 83, 76, 75, 74]}
+{"id": "linear-algebra_T3_1_251", "domain": "linear-algebra", "type": "T3_path", "query": "What is the prerequisite chain from Scalar to Image Tensor?", "ground_truth": ["Image Tensor", "Tensor", "N-Dimensional Vector", "Vector", "Scalar"], "concept_id": 251, "hop_depth": 4, "path_ids": [251, 213, 6, 2, 1]}
+{"id": "linear-algebra_T3_1_67", "domain": "linear-algebra", "type": "T3_path", "query": "What is the prerequisite chain from Scalar to Unique Solution?", "ground_truth": ["Unique Solution", "Solution Set", "System of Equations", "Linear Equation", "Scalar"], "concept_id": 67, "hop_depth": 4, "path_ids": [67, 66, 52, 51, 1]}
+{"id": "linear-algebra_T3_1_27", "domain": "linear-algebra", "type": "T3_path", "query": "What is the prerequisite chain from Scalar to Dimension of Space?", "ground_truth": ["Dimension of Space", "Vector Space", "Linear Combination", "Scalar Multiplication", "Scalar"], "concept_id": 27, "hop_depth": 4, "path_ids": [27, 26, 19, 8, 1]}
+{"id": "linear-algebra_T3_1_50", "domain": "linear-algebra", "type": "T3_path", "query": "What is the prerequisite chain from Scalar to Block Matrix?", "ground_truth": ["Block Matrix", "Matrix", "Vector", "Scalar"], "concept_id": 50, "hop_depth": 3, "path_ids": [50, 28, 2, 1]}
+{"id": "linear-algebra_T3_1_69", "domain": "linear-algebra", "type": "T3_path", "query": "What is the prerequisite chain from Scalar to No Solution?", "ground_truth": ["No Solution", "Solution Set", "System of Equations", "Linear Equation", "Scalar"], "concept_id": 69, "hop_depth": 4, "path_ids": [69, 66, 52, 51, 1]}
+{"id": "linear-algebra_T3_1_283", "domain": "linear-algebra", "type": "T3_path", "query": "What is the prerequisite chain from Scalar to Sensor Fusion?", "ground_truth": ["Sensor Fusion", "Kalman Filter", "State Vector", "Vector", "Scalar"], "concept_id": 283, "hop_depth": 4, "path_ids": [283, 284, 285, 2, 1]}
+{"id": "linear-algebra_T3_74_192", "domain": "linear-algebra", "type": "T3_path", "query": "What is the prerequisite chain from Function to ReLU?", "ground_truth": ["ReLU", "Activation Function", "Function"], "concept_id": 192, "hop_depth": 2, "path_ids": [192, 191, 74]}
+{"id": "linear-algebra_T3_1_108", "domain": "linear-algebra", "type": "T3_path", "query": "What is the prerequisite chain from Scalar to Multiplicative Property?", "ground_truth": ["Multiplicative Property", "Determinant", "Matrix", "Vector", "Scalar"], "concept_id": 108, "hop_depth": 4, "path_ids": [108, 101, 28, 2, 1]}
+{"id": "linear-algebra_T3_1_149", "domain": "linear-algebra", "type": "T3_path", "query": "What is the prerequisite chain from Scalar to Condition Number?", "ground_truth": ["Condition Number", "Matrix Inverse", "Matrix Multiplication", "Matrix-Vector Product", "Matrix Scalar Multiply", "Scalar"], "concept_id": 149, "hop_depth": 5, "path_ids": [149, 46, 37, 36, 35, 1]}
+{"id": "linear-algebra_T3_1_247", "domain": "linear-algebra", "type": "T3_path", "query": "What is the prerequisite chain from Scalar to KKT Conditions?", "ground_truth": ["KKT Conditions", "Lagrange Multiplier", "Gradient Vector", "N-Dimensional Vector", "Vector", "Scalar"], "concept_id": 247, "hop_depth": 5, "path_ids": [247, 245, 185, 6, 2, 1]}
+{"id": "linear-algebra_T3_1_247", "domain": "linear-algebra", "type": "T3_path", "query": "What is the prerequisite chain from Scalar to KKT Conditions?", "ground_truth": ["KKT Conditions", "Lagrange Multiplier", "Gradient Vector", "N-Dimensional Vector", "Vector", "Scalar"], "concept_id": 247, "hop_depth": 5, "path_ids": [247, 245, 185, 6, 2, 1]}
+{"id": "linear-algebra_T3_1_9", "domain": "linear-algebra", "type": "T3_path", "query": "What is the prerequisite chain from Scalar to Vector Subtraction?", "ground_truth": ["Vector Subtraction", "Vector Addition", "Vector", "Scalar"], "concept_id": 9, "hop_depth": 3, "path_ids": [9, 7, 2, 1]}
+{"id": "linear-algebra_T3_1_282", "domain": "linear-algebra", "type": "T3_path", "query": "What is the prerequisite chain from Scalar to Camera Calibration?", "ground_truth": ["Camera Calibration", "Camera Matrix", "Matrix", "Vector", "Scalar"], "concept_id": 282, "hop_depth": 4, "path_ids": [282, 272, 28, 2, 1]}
+{"id": "linear-algebra_T4_FOUND", "domain": "linear-algebra", "type": "T4_aggregate", "query": "List all FOUND concepts in this knowledge graph", "ground_truth": ["Scalar", "Vector", "Vector Notation", "2D Vector", "3D Vector", "N-Dimensional Vector", "Vector Addition", "Scalar Multiplication", "Vector Subtraction", "Dot Product", "Cross Product", "Vector Magnitude", "Unit Vector", "Vector Normalization", "Euclidean Distance", "L1 Norm", "L2 Norm", "L-Infinity Norm", "Linear Combination", "Span", "Linear Independence", "Linear Dependence", "Basis Vector", "Standard Basis", "Coordinate System", "Vector Space", "Dimension of Space"], "taxonomy_id": "FOUND", "hop_depth": 0}
+{"id": "linear-algebra_T4_MATOP", "domain": "linear-algebra", "type": "T4_aggregate", "query": "List all MATOP concepts in this knowledge graph", "ground_truth": ["Matrix", "Matrix Notation", "Matrix Dimensions", "Row Vector", "Column Vector", "Matrix Entry", "Matrix Addition", "Matrix Scalar Multiply", "Matrix-Vector Product", "Matrix Multiplication", "Matrix Transpose", "Symmetric Matrix", "Identity Matrix", "Diagonal Matrix", "Triangular Matrix", "Upper Triangular", "Lower Triangular", "Orthogonal Matrix", "Matrix Inverse", "Invertible Matrix", "Sparse Matrix", "Dense Matrix", "Block Matrix"], "taxonomy_id": "MATOP", "hop_depth": 0}
+{"id": "linear-algebra_T4_LINSYS", "domain": "linear-algebra", "type": "T4_aggregate", "query": "List all LINSYS concepts in this knowledge graph", "ground_truth": ["Linear Equation", "System of Equations", "Matrix Equation Form", "Augmented Matrix", "Gaussian Elimination", "Row Operations", "Row Swap", "Row Scaling", "Row Addition", "Row Echelon Form", "Reduced Row Echelon Form", "Pivot Position", "Pivot Column", "Free Variable", "Basic Variable", "Solution Set", "Unique Solution", "Infinite Solutions", "No Solution", "Homogeneous System", "Trivial Solution", "Numerical Stability", "Back Substitution"], "taxonomy_id": "LINSYS", "hop_depth": 0}
+{"id": "linear-algebra_T4_TRANS", "domain": "linear-algebra", "type": "T4_aggregate", "query": "List all TRANS concepts in this knowledge graph", "ground_truth": ["Function", "Linear Transformation", "Transformation Matrix", "Domain", "Codomain", "Image", "Rotation Matrix", "2D Rotation", "3D Rotation", "Scaling Matrix", "Uniform Scaling", "Non-Uniform Scaling", "Shear Matrix", "Reflection Matrix", "Projection", "Orthogonal Projection", "Composition of Transforms", "Kernel", "Null Space", "Range", "Column Space", "Rank", "Nullity", "Rank-Nullity Theorem", "Invertible Transform", "Change of Basis", "Basis Transition Matrix"], "taxonomy_id": "TRANS", "hop_depth": 0}
+{"id": "linear-algebra_T4_DETERM", "domain": "linear-algebra", "type": "T4_aggregate", "query": "List all DETERM concepts in this knowledge graph", "ground_truth": ["Determinant", "2x2 Determinant", "3x3 Determinant", "Cofactor Expansion", "Minor", "Cofactor", "Determinant Properties", "Multiplicative Property", "Transpose Determinant", "Singular Matrix", "Volume Scaling Factor", "Signed Area", "Cramers Rule"], "taxonomy_id": "DETERM", "hop_depth": 0}
+{"id": "linear-algebra_T4_EIGEN", "domain": "linear-algebra", "type": "T4_aggregate", "query": "List all EIGEN concepts in this knowledge graph", "ground_truth": ["Eigenvalue", "Eigenvector", "Eigen Equation", "Characteristic Polynomial", "Characteristic Equation", "Eigenspace", "Algebraic Multiplicity", "Geometric Multiplicity", "Diagonalization", "Diagonal Form", "Similar Matrices", "Complex Eigenvalue", "Spectral Theorem", "Symmetric Eigenvalues", "Power Iteration", "Dominant Eigenvalue", "Eigendecomposition"], "taxonomy_id": "EIGEN", "hop_depth": 0}
+{"id": "linear-algebra_T4_DECOMP", "domain": "linear-algebra", "type": "T4_aggregate", "query": "List all DECOMP concepts in this knowledge graph", "ground_truth": ["Matrix Factorization", "LU Decomposition", "Partial Pivoting", "QR Decomposition", "Gram-Schmidt QR", "Householder QR", "Cholesky Decomposition", "Positive Definite Matrix", "SVD", "Singular Value", "Left Singular Vector", "Right Singular Vector", "Full SVD", "Compact SVD", "Truncated SVD", "Low-Rank Approximation", "Matrix Rank", "Numerical Rank", "Condition Number"], "taxonomy_id": "DECOMP", "hop_depth": 0}
+{"id": "linear-algebra_T4_INPROD", "domain": "linear-algebra", "type": "T4_aggregate", "query": "List all INPROD concepts in this knowledge graph", "ground_truth": ["Abstract Vector Space", "Subspace", "Vector Space Axioms", "Inner Product", "Inner Product Space", "Norm from Inner Product", "Cauchy-Schwarz Inequality", "Orthogonality", "Orthogonal Vectors", "Orthonormal Set", "Orthonormal Basis", "Gram-Schmidt Process", "Projection onto Subspace", "Least Squares Problem", "Normal Equations", "Row Space", "Left Null Space", "Four Subspaces", "Pseudoinverse"], "taxonomy_id": "INPROD", "hop_depth": 0}
+{"id": "linear-algebra_T4_MLBASE", "domain": "linear-algebra", "type": "T4_aggregate", "query": "List all MLBASE concepts in this knowledge graph", "ground_truth": ["Feature Vector", "Feature Matrix", "Data Matrix", "Covariance Matrix", "Correlation Matrix", "Standardization", "PCA", "Principal Component", "Variance Explained", "Scree Plot", "Dimensionality Reduction", "Linear Regression", "Design Matrix", "Ridge Regression", "Lasso Regression", "Regularization", "Gradient Vector", "Gradient Descent", "Batch Gradient Descent", "Learning Rate"], "taxonomy_id": "MLBASE", "hop_depth": 0}
+{"id": "linear-algebra_T4_NEURAL", "domain": "linear-algebra", "type": "T4_aggregate", "query": "List all NEURAL concepts in this knowledge graph", "ground_truth": ["Perceptron", "Neuron Model", "Activation Function", "ReLU", "Sigmoid", "Tanh", "Softmax", "Weight Matrix", "Bias Vector", "Forward Propagation", "Backpropagation", "Chain Rule Matrices", "Loss Function", "Cross-Entropy Loss", "Neural Network Layer", "Hidden Layer", "Deep Network", "Convolutional Layer", "Convolution Kernel", "Stride", "Padding", "Pooling Layer", "Batch Normalization", "Layer Normalization", "Tensor", "Tensor Operations"], "taxonomy_id": "NEURAL", "hop_depth": 0}
+{"id": "linear-algebra_T4_GENAI", "domain": "linear-algebra", "type": "T4_aggregate", "query": "List all GENAI concepts in this knowledge graph", "ground_truth": ["Embedding", "Embedding Space", "Word Embedding", "Semantic Similarity", "Cosine Similarity", "Attention Mechanism", "Self-Attention", "Cross-Attention", "Query Matrix", "Key Matrix", "Value Matrix", "Attention Score", "Attention Weights", "Multi-Head Attention", "Transformer Architecture", "Position Encoding", "LoRA", "Latent Space", "Interpolation"], "taxonomy_id": "GENAI", "hop_depth": 0}
+{"id": "linear-algebra_T4_OPTIM", "domain": "linear-algebra", "type": "T4_aggregate", "query": "List all OPTIM concepts in this knowledge graph", "ground_truth": ["Hessian Matrix", "Convexity", "Convex Function", "Newtons Method", "Quasi-Newton Method", "BFGS Algorithm", "SGD", "Mini-Batch SGD", "Momentum", "Adam Optimizer", "RMSprop", "Lagrange Multiplier", "Constrained Optimization", "KKT Conditions"], "taxonomy_id": "OPTIM", "hop_depth": 0}
+{"id": "linear-algebra_T4_IMGPROC", "domain": "linear-algebra", "type": "T4_aggregate", "query": "List all IMGPROC concepts in this knowledge graph", "ground_truth": ["Image Matrix", "Grayscale Image", "RGB Image", "Image Tensor", "Image Convolution", "Image Filter", "Blur Filter", "Sharpen Filter", "Edge Detection", "Sobel Operator", "Fourier Transform", "Frequency Domain", "Image Compression", "Color Space Transform", "Feature Detection", "Homography"], "taxonomy_id": "IMGPROC", "hop_depth": 0}
+{"id": "linear-algebra_T4_GEOM3D", "domain": "linear-algebra", "type": "T4_aggregate", "query": "List all GEOM3D concepts in this knowledge graph", "ground_truth": ["3D Coordinate System", "Euler Angles", "Gimbal Lock", "Quaternion", "Quaternion Rotation", "Homogeneous Coordinates", "Rigid Body Transform", "SE3 Transform", "Camera Matrix", "Intrinsic Parameters", "Extrinsic Parameters", "Projection Matrix", "Perspective Projection", "Stereo Vision", "Triangulation", "Epipolar Geometry", "Point Cloud"], "taxonomy_id": "GEOM3D", "hop_depth": 0}
+{"id": "linear-algebra_T4_AUTON", "domain": "linear-algebra", "type": "T4_aggregate", "query": "List all AUTON concepts in this knowledge graph", "ground_truth": ["LIDAR Point Cloud", "Camera Calibration", "Sensor Fusion", "Kalman Filter", "State Vector", "Measurement Vector", "Prediction Step", "Update Step", "Kalman Gain", "Extended Kalman Filter", "State Estimation", "SLAM", "Localization", "Mapping", "Object Detection", "Object Tracking", "Bounding Box", "Path Planning", "Motion Planning", "Trajectory Optimization"], "taxonomy_id": "AUTON", "hop_depth": 0}
+{"id": "linear-algebra_T5_189_190", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Perceptron relate to Neuron Model?", "ground_truth": ["Perceptron", "Neuron Model"], "concept_id_a": 189, "concept_id_b": 190, "hop_depth": 1}
+{"id": "linear-algebra_T5_19_7", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Linear Combination relate to Vector Addition?", "ground_truth": ["Linear Combination", "Vector Addition"], "concept_id_a": 19, "concept_id_b": 7, "hop_depth": 1}
+{"id": "linear-algebra_T5_70_53", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Homogeneous System relate to Matrix Equation Form?", "ground_truth": ["Homogeneous System", "Matrix Equation Form", "System of Equations"], "concept_id_a": 70, "concept_id_b": 53, "hop_depth": 1}
+{"id": "linear-algebra_T5_21_20", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Linear Independence relate to Span?", "ground_truth": ["Linear Independence", "Span", "Linear Combination"], "concept_id_a": 21, "concept_id_b": 20, "hop_depth": 1}
+{"id": "linear-algebra_T5_273_272", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Intrinsic Parameters relate to Camera Matrix?", "ground_truth": ["Intrinsic Parameters", "Camera Matrix"], "concept_id_a": 273, "concept_id_b": 272, "hop_depth": 1}
+{"id": "linear-algebra_T5_11_5", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Cross Product relate to 3D Vector?", "ground_truth": ["Cross Product", "3D Vector"], "concept_id_a": 11, "concept_id_b": 5, "hop_depth": 1}
+{"id": "linear-algebra_T5_259_258", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Frequency Domain relate to Fourier Transform?", "ground_truth": ["Frequency Domain", "Fourier Transform"], "concept_id_a": 259, "concept_id_b": 258, "hop_depth": 1}
+{"id": "linear-algebra_T5_102_101", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does 2x2 Determinant relate to Determinant?", "ground_truth": ["2x2 Determinant", "Determinant"], "concept_id_a": 102, "concept_id_b": 101, "hop_depth": 1}
+{"id": "linear-algebra_T5_23_20", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Basis Vector relate to Span?", "ground_truth": ["Basis Vector", "Span"], "concept_id_a": 23, "concept_id_b": 20, "hop_depth": 1}
+{"id": "linear-algebra_T5_155_12", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Norm from Inner Product relate to Vector Magnitude?", "ground_truth": ["Norm from Inner Product", "Vector Magnitude"], "concept_id_a": 155, "concept_id_b": 12, "hop_depth": 1}
+{"id": "linear-algebra_T5_73_60", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Back Substitution relate to Row Echelon Form?", "ground_truth": ["Back Substitution", "Row Echelon Form"], "concept_id_a": 73, "concept_id_b": 60, "hop_depth": 1}
+{"id": "linear-algebra_T5_86_76", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Shear Matrix relate to Transformation Matrix?", "ground_truth": ["Shear Matrix", "Transformation Matrix"], "concept_id_a": 86, "concept_id_b": 76, "hop_depth": 1}
+{"id": "linear-algebra_T5_200_199", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Chain Rule Matrices relate to Backpropagation?", "ground_truth": ["Chain Rule Matrices", "Backpropagation"], "concept_id_a": 200, "concept_id_b": 199, "hop_depth": 1}
+{"id": "linear-algebra_T5_145_144", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Truncated SVD relate to Compact SVD?", "ground_truth": ["Truncated SVD", "Compact SVD"], "concept_id_a": 145, "concept_id_b": 144, "hop_depth": 1}
+{"id": "linear-algebra_T5_65_64", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Basic Variable relate to Free Variable?", "ground_truth": ["Basic Variable", "Free Variable", "Reduced Row Echelon Form"], "concept_id_a": 65, "concept_id_b": 64, "hop_depth": 1}
+{"id": "linear-algebra_T5_163_162", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Least Squares Problem relate to Projection onto Subspace?", "ground_truth": ["Least Squares Problem", "Projection onto Subspace"], "concept_id_a": 163, "concept_id_b": 162, "hop_depth": 1}
+{"id": "linear-algebra_T5_39_38", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Symmetric Matrix relate to Matrix Transpose?", "ground_truth": ["Symmetric Matrix", "Matrix Transpose"], "concept_id_a": 39, "concept_id_b": 38, "hop_depth": 1}
+{"id": "linear-algebra_T5_217_215", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Word Embedding relate to Embedding?", "ground_truth": ["Word Embedding", "Embedding"], "concept_id_a": 217, "concept_id_b": 215, "hop_depth": 1}
+{"id": "linear-algebra_T5_282_272", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Camera Calibration relate to Camera Matrix?", "ground_truth": ["Camera Calibration", "Camera Matrix"], "concept_id_a": 282, "concept_id_b": 272, "hop_depth": 1}
+{"id": "linear-algebra_T5_265_264", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Euler Angles relate to 3D Coordinate System?", "ground_truth": ["Euler Angles", "3D Coordinate System"], "concept_id_a": 265, "concept_id_b": 264, "hop_depth": 1}
+{"id": "linear-algebra_T5_170_28", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Feature Matrix relate to Matrix?", "ground_truth": ["Feature Matrix", "Matrix"], "concept_id_a": 170, "concept_id_b": 28, "hop_depth": 1}
+{"id": "linear-algebra_T5_172_39", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Covariance Matrix relate to Symmetric Matrix?", "ground_truth": ["Covariance Matrix", "Symmetric Matrix"], "concept_id_a": 172, "concept_id_b": 39, "hop_depth": 1}
+{"id": "linear-algebra_T5_142_139", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Right Singular Vector relate to SVD?", "ground_truth": ["Right Singular Vector", "SVD"], "concept_id_a": 142, "concept_id_b": 139, "hop_depth": 1}
+{"id": "linear-algebra_T5_75_74", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Linear Transformation relate to Function?", "ground_truth": ["Linear Transformation", "Function"], "concept_id_a": 75, "concept_id_b": 74, "hop_depth": 1}
+{"id": "linear-algebra_T5_233_232", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Interpolation relate to Latent Space?", "ground_truth": ["Interpolation", "Latent Space"], "concept_id_a": 233, "concept_id_b": 232, "hop_depth": 1}
+{"id": "linear-algebra_T5_242_186", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Momentum relate to Gradient Descent?", "ground_truth": ["Momentum", "Gradient Descent"], "concept_id_a": 242, "concept_id_b": 186, "hop_depth": 1}
+{"id": "linear-algebra_T5_124_122", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Similar Matrices relate to Diagonalization?", "ground_truth": ["Similar Matrices", "Diagonalization"], "concept_id_a": 124, "concept_id_b": 122, "hop_depth": 1}
+{"id": "linear-algebra_T5_56_55", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Row Operations relate to Gaussian Elimination?", "ground_truth": ["Row Operations", "Gaussian Elimination", "Augmented Matrix"], "concept_id_a": 56, "concept_id_b": 55, "hop_depth": 1}
+{"id": "linear-algebra_T5_30_28", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Matrix Dimensions relate to Matrix?", "ground_truth": ["Matrix Dimensions", "Matrix"], "concept_id_a": 30, "concept_id_b": 28, "hop_depth": 1}
+{"id": "linear-algebra_T5_30_29", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Matrix Dimensions relate to Matrix Notation?", "ground_truth": ["Matrix Dimensions", "Matrix Notation", "Matrix"], "concept_id_a": 30, "concept_id_b": 29, "hop_depth": 1}
+{"id": "linear-algebra_T5_198_196", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Forward Propagation relate to Weight Matrix?", "ground_truth": ["Forward Propagation", "Weight Matrix"], "concept_id_a": 198, "concept_id_b": 196, "hop_depth": 1}
+{"id": "linear-algebra_T5_131_46", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Matrix Factorization relate to Matrix Inverse?", "ground_truth": ["Matrix Factorization", "Matrix Inverse", "Matrix Multiplication"], "concept_id_a": 131, "concept_id_b": 46, "hop_depth": 1}
+{"id": "linear-algebra_T5_110_47", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Singular Matrix relate to Invertible Matrix?", "ground_truth": ["Singular Matrix", "Invertible Matrix"], "concept_id_a": 110, "concept_id_b": 47, "hop_depth": 1}
+{"id": "linear-algebra_T5_129_128", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Dominant Eigenvalue relate to Power Iteration?", "ground_truth": ["Dominant Eigenvalue", "Power Iteration"], "concept_id_a": 129, "concept_id_b": 128, "hop_depth": 1}
+{"id": "linear-algebra_T5_126_39", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Spectral Theorem relate to Symmetric Matrix?", "ground_truth": ["Spectral Theorem", "Symmetric Matrix"], "concept_id_a": 126, "concept_id_b": 39, "hop_depth": 1}
+{"id": "linear-algebra_T5_140_139", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Singular Value relate to SVD?", "ground_truth": ["Singular Value", "SVD"], "concept_id_a": 140, "concept_id_b": 139, "hop_depth": 1}
+{"id": "linear-algebra_T5_260_248", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Image Compression relate to Image Matrix?", "ground_truth": ["Image Compression", "Image Matrix"], "concept_id_a": 260, "concept_id_b": 248, "hop_depth": 1}
+{"id": "linear-algebra_T5_219_13", "domain": "linear-algebra", "type": "T5_cross_concept", "query": "How does Cosine Similarity relate to Unit Vector?", "ground_truth": ["Cosine Similarity", "Unit Vector"], "concept_id_a": 219, "concept_id_b": 13, "hop_depth": 1}