{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"derivations","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":{"q":"mko-matrix"},"kind":"guides","total":2,"pages":1,"page":1,"entries":[{"id":"derivation-mko-matrix-01","title":{"zh-Hant":"矩陣：尺寸、一起算索引、逐項加法與轉置","en":"Matrices: dimensions, one-based indexing, entrywise addition and transpose"},"summary":{"zh-Hant":"設 M=[[2,−1,0],[5,3,7]]、N=[[1,4,−2],[0,−3,6]]（每個內層括號是一行，row）。求兩者尺寸，讀出 M(1,2)、M(2,3)，判定 M(3,1) 是否存在；求 M+N、Mᵀ、Nᵀ；對任意同尺寸實矩陣證明 (A+B)ᵀ=Aᵀ+Bᵀ 與 (Aᵀ)ᵀ=A，並在 M、N 上核對；說明 M+Mᵀ 無定義；最後以反例區分轉置與「逐行倒序」及「反矩陣」，並把程式的零起算索引對應回一起算索引。","en":"Let M=[[2,−1,0],[5,3,7]] and N=[[1,4,−2],[0,−3,6]] (each inner bracket is one row). Find their dimensions, read off M(1,2) and M(2,3), and decide whether M(3,1) exists; compute M+N, Mᵀ and Nᵀ; prove (A+B)ᵀ=Aᵀ+Bᵀ and (Aᵀ)ᵀ=A for all real matrices of equal size and check them on M and N; explain why M+Mᵀ is undefined; finally use counterexamples to separate the transpose from reversing the rows and from the inverse matrix, and translate zero-based program indices back to one-based indices."},"mko_ids":["mko-matrix"]},{"id":"derivation-mko-matrix-rank-01","title":{"zh-Hant":"三階矩陣的完整RREF與上下界","en":"Full RREF and rank bounds"},"summary":{"zh-Hant":"A=[[1,2,3],[2,4,6],[1,1,1]]的row與column rank均為2。","en":"A=[[1,2,3],[2,4,6],[1,1,1]] has row and column rank 2."},"mko_ids":["mko-matrix-rank"]}],"title":"概念推導導讀"}