{"schema_version":"ocme-site-results-v1","locale":"en","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-general-linear-map"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-general-linear-map-01","title":{"zh-Hant":"實向量空間間線性映射：矩陣映射的全稱證書與仿射、黑箱反例","en":"Linear maps between real vector spaces: a universal certificate for a matrix map, with affine and black-box counterexamples"},"summary":{"zh-Hant":"設 A 為第一行 (2,−1,0)、第二行 (1,3,4) 的 2×3 整數矩陣，T:ℝ³→ℝ²，T(x)=Ax。（一）證明對所有 u,v∈ℝ³ 與所有實數 α,β，T(αu+βv)=αT(u)+βT(v)。（二）證明 S(x)=Ax+b（b=(1,0)）不是線性映射，並證明對任意 b∈ℝ²，x↦Ax+b 線性當且僅當 b=0。（三）說明黑箱 F(x)=Ax+(x₁²−x₁,0) 通過一個樣本測試，卻不是線性映射。","en":"Let A be the 2-by-3 integer matrix with first row (2,−1,0) and second row (1,3,4), and let T:ℝ³→ℝ², T(x)=Ax. (1) Prove that T(αu+βv)=αT(u)+βT(v) for all u,v∈ℝ³ and all reals α,β. (2) Prove that S(x)=Ax+b with b=(1,0) is not linear, and that for every b∈ℝ² the map x↦Ax+b is linear if and only if b=0. (3) Show that the black box F(x)=Ax+(x₁²−x₁,0) passes one sample test and yet is not linear."},"mko_ids":["mko-general-linear-map"]}],"title":"Concept derivation guides"}