{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"paths","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{},"path":{"id":"path-pulse-foundations-20261007","title":{"zh-Hant":"矩陣、實二階線性代數與標記三角形、座標變換：七輪脈衝","en":"Matrices, Real 2×2 Linear Algebra, Labeled Triangles, and Coordinate Transformations: Seven-Round Workflow"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-vector","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quadratic-equation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-triangle","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-distance-formula","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-ratio-proportion","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-power-root","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-coordinate-plane","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-matrix","position":2,"requirement":"optional","rationale":{"zh-Hant":"矩陣以行、列組織元素；本輪使用非空有限實矩陣及精確整數陣列切片。","en":"Matrices organize entries by rows and columns; this pulse uses nonempty finite real matrices and exact integer-array instances."}},{"id":"mko-linear-transformation","position":3,"requirement":"optional","rationale":{"zh-Hant":"先在 R²→R² 的矩陣切片驗證加法與純量相容，不把帶非零平移的仿射式叫線性映射。","en":"Use the explicit R²→R² matrix model for additivity and scalar compatibility; an affine expression with nonzero translation is not linear."}},{"id":"mko-determinant","position":4,"requirement":"optional","rationale":{"zh-Hant":"2×2實矩陣的 ad−bc、非零行列式可逆性、乘積與純量縮放切片。","en":"This entry covers ad−bc, invertibility over R, product determinants, and scalar scaling for 2×2 real matrices."}},{"id":"mko-eigenvalue","position":5,"requirement":"optional","rationale":{"zh-Hant":"特徵向量須非零；候選特徵空間核含零向量，實二階矩陣可沒有實特徵值。","en":"An eigenvector must be nonzero, whereas a candidate-scalar kernel includes zero; a real 2×2 matrix may have no real eigenvalues."}},{"id":"mko-congruence","position":6,"requirement":"optional","rationale":{"zh-Hant":"限定正交等單位平面中的標記非退化三角形，以三組對應邊長判全等，允許鏡射。","en":"In an equal-unit orthonormal plane, this slice compares all three corresponding side lengths of labeled nondegenerate triangles and allows reflection."}},{"id":"mko-similarity","position":7,"requirement":"optional","rationale":{"zh-Hant":"指定由A到B的正長度倍率 k，平方長度與指定面積以 k² 縮放，不把負座標係數當負長度倍率。","en":"Specify the positive length scale k from A to B; squared lengths and a given area scale by k². A negative coordinate coefficient is not a negative length scale."}},{"id":"mko-geometric-transformations","position":8,"requirement":"optional","rationale":{"zh-Hant":"平移、原點逆時針四分之一圈的整數倍旋轉、指定軸鏡射是全平面等距；均勻縮放先分清零、±1與其他係數。","en":"Translations, counterclockwise quarter-turn rotations about the origin, and specified line reflections are plane isometries; uniform scaling distinguishes zero, ±1, and other coefficients."}}]},"title":"矩陣、實二階線性代數與標記三角形、座標變換：七輪脈衝"}