{"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-linear-independence-basis"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-linear-independence-basis-01","title":{"zh-Hant":"線性獨立與基底：以精確消去證明 ℝ² 的一組有序基底，並以兩個反例分開兩條責任","en":"Linear independence and bases: proving an ordered basis of ℝ² by exact elimination, with two counterexamples that separate the two obligations"},"summary":{"zh-Hant":"在 ℝ² 中令 v_1=(1,2)、v_2=(3,5)、v_3=(1,1)、w=(4,7)。(1) 以精確消去證明 (v_1,v_2) 線性獨立；(2) 對任意 (p,q) 求出係數，證明 span(v_1,v_2)=ℝ²，因而 (v_1,v_2) 是有序基底；(3) 求 w 的唯一座標；(4) 證明 (v_1) 獨立但不生成 ℝ²，而 (v_1,v_2,v_3) 生成 ℝ² 但不獨立，且 w 在其中的表示不唯一。","en":"In ℝ² let v_1=(1,2), v_2=(3,5), v_3=(1,1), w=(4,7). (1) Prove by exact elimination that (v_1,v_2) is linearly independent; (2) find the coefficients for an arbitrary (p,q), proving span(v_1,v_2)=ℝ², so that (v_1,v_2) is an ordered basis; (3) find the unique coordinates of w; (4) prove that (v_1) is independent but does not span ℝ², while (v_1,v_2,v_3) spans ℝ² but is not independent, and w has more than one representation in it."},"mko_ids":["mko-linear-independence-basis"]}],"title":"概念推導導讀"}